It’s often considered, in the theater and elsewhere, to be bad luck to wish someone “good luck.” Instead, some people say, “Break a leg!” This appears to be from the German “Brechen sie Hals und Beine!” and “Hals und Beinbruch.”
   
“One cannot offend a German hunter more than by wishing him good luck; the orthodox formula when parting with him is: ‘Weidmanns Heil! Brechen sie Hals und Beine!’” was printed in Notes and Queries in 1906. “‘Hals und beinbruch,,’—or ‘hope you break your neck and legs’—which is a greeting customary among German fliers when wishing each other good luck” was printed in the Lima (OH) Star on August 12, 1927. “I did not wish a comrade ‘Good luck!’ before he took off, but ‘Hals und Beinbruch!’ (May you break your neck and your limbs!)” was printed in the book War Flying in Macedonia (1935) by Haupt Heydemarck.
   
The expression probably came into the theater from German opera stars. “Lauritz Melchior’s wife always says to him ‘Hals und Beinbruch” (I hope you break your arms and legs) when he goes onstage to sing, because to wish an opera star ‘Good luck’ is considered the worst of all possible omens” was printed in the Daily News (New York, NY) on February 1, 1940, in the column “Broadway” by Danton Walker.
   
“Why do we tell actors to ‘break a leg’?”/“Because every play has a cast” is a joke on the saying. “Fracture a femur” is an occasionally used jocular variation of “break a leg.” “Merde!” and “Toi, toi, toi” are other good luck sayings.
 
     
Wiktionary: break a leg
Etymology
Unknown; many unproven and widely debated theories exist. One of the most plausible is that it comes from Yiddish הצלחה און ברכה (hatsloche un broche) through the heavy Yiddish influence in the American theater, from German Hals- und Beinbruch (“neck and leg break”), perhaps a corruption of the Hebrew הַצְלָחָה וּבְרָכָה (hatslachá uvrachá, “success and blessing”).
Verb
break a leg (imperative only)
1.(idiomatic) To perform well in a theatrical production or comparable endeavor.
 
Wikipedia: Break a leg
“Break a leg” is a well-known idiom in theatre which means “good luck.” It is typically said to actors and musicians before they go on stage to perform. The origin of the phrase is obscure.
 
The expression reflects a theatrical superstition in which wishing a person “good luck” is considered bad luck. The expression is sometimes used outside the theatre as superstitions and customs travel through other professions and then into common use. Among professional dancers, the phrase “break a leg” is replaced with “merde”.
 
Origins
The earliest known example in print is from Edna Ferber’s 1939 A Peculiar Treasure in which she writes about the fascination of the theater, “...and all the understudies sitting in the back row politely wishing the various principals would break a leg”. In Bernard Sobel’s 1948 The Theatre Handbook and Digest of Plays, he writes about theatrical superstitions: “...before a performance actors never wish each other good luck, but say ‘I hope you break a leg.’” There is anecdotal evidence from theatrical memoirs and personal letters as early as the 1920s.
 
There are several theories behind the origin of the phrase. Few are supported by contemporary writings. The theories listed below are some of the more popular explanations.
 
Opposite meaning
People in theatre consider it bad luck to wish an actor good luck, so instead they wish the opposite, by saying “break a leg!”.
 
On October 1, 1921 in the New Statesman, a British liberal political and cultural magazine, an article was published, “A Defence of Superstition”, written by urbane Irish nationalist Robert Wilson Lynd. Lynd said that the theatre was the second-most superstitious institution in England, after horse racing. In horse racing, Lynd asserted, to wish a man luck is considered unlucky, so “You should say something insulting such as, ‘May you break your leg!’” Lynd did not attribute the phrase in any way to theatre people, though he was familiar with many of them.
         
Google Books
Notes and Queries
1906
Pg. 231:
One cannot offend a German hunter more than by wishing him good luck; the orthodox formula when parting with him is: “Weidmanns Heil! Brechen sie Hals und Beine!”
G. KRUEGER
 
Newspapers.com
12 August 1927, Lima (OH) Star, pg. 1, col. 8:
German Flier Is
Set For Take-off
To United States
BERLIN, Aug. 11.—(AP)—
(...)
During the afternoon numerous German pilots visited Keonnecke to express their fervent wishes for his success. Among them was Ernest Udet, war ace, who with his inseparable companion dog Bobby, stepped up and said, “Hals und beinbruch,”—or “hope you break your neck and legs”—which is a greeting customary among German fliers when wishing each other good luck.
   
Google Books
War Flying in Macedonia
By Haupt Heydemarck
London, UK: J. Hamilton Ltd.
1935
Pg. 122:
I did not wish a comrade “Good luck!” before he took off, but “Hals und Beinbruch!” (May you break your neck and your limbs!)
   
Google Boks
Department of Science, Art and Literature: Hearings Before the Committee on Patents, House of Representatives
United States. Congress. House. Committee on Patents
June 20, 1935
Pg. 25:
STATEMENT OF MME. LILLIAN EVANTI, OPERA AND CONCERT ARTIST, OF WASHINGTON, D. C.
Pg. 26:
I still hear ringing in my ears the fine words of encouragement with each wishing me “good luck.” I am reminded of how the Germans wish you good luck with “Hals und Beinbruch”, which means “Fall down and break your neck and legs”; or, as the Italians say “In boca lupo”, which means “into the mouth o the wolf.”
 
22 February 1936, New York (NY) Times, “Letters to the Sports Editor” ‘Bad Luck’ Expressions,” Sports sec., pg. 11, col. 2:
It is a custom in most German-speaking countries never to wish anybody good luck so far as sports is concerned. A common expression is “Hals und Beinbruch,” implying “I hope you break your neck and legs.”
(...)
PAUL HOFMEISTER.
Astoria, L. I., Feb. 17, 1936.
   
Google Books
Backstage at the Opera
By Rose Heylbut and Aimé Gerber
New York, NY: Thomas Y. Crowell
1937
Pg. 127:
The German contingent of singers believes it bad luck to express a wish for “good luck” to a performer on his way to the stage. (...) Thus, the ritual is to wish the departing one “Hals und Bein Bruch” which is translated as a breaking of the neck and the limbs.
 
Google Books
A Peculiar Treasure:
An Autobiography
By Edna Ferber
New York, NY: Doran & Company
1939
Pg. 354:
And when that grisly night of dress rehearsal finally comes round, and strange figures enter the dim auditorium and grope for seats and whisper and mumble and creep about and you make out the dressmaker and the dressmaker’s assistant and the girl from Berdorfs (the star’s clothes) and the girl from Saks’ (the ingenue’s) and the friend of management, and somebody’s uncle, and all the understudies sitting in the back row politely wishing the various principals would break a leg—it is then that everything goes suddenly completely and inextricably wrong and you realize that tomorrow night is just twenty-four hours away.
   
Google Books
August 1939, Town & Country (New York, NY), “Dear General—what a surprise!” by Ludwig Bemelmans, pg. 61, col. 3:
EVERYBODY wished every one the good luck. The slim German girl who played the heroine spit three times—toit! toi! toi!—on my table collar, the Continental actors’ way of wishing good luck. She also said, “Hals und Beinbruch!”—a phrase that skiers and acrobats shout to each other.
     
Newspapers.com
23 January 1940, Kansas City (MO) Times, “Times Describes Lauritz Melchior at Home on a Merry Evening,” pg. 16, col. 4:
“At the murmuring strains of Wagner’s prelude, (Laurite—ed.) Melchior throws away his cigar and clears his throat. Kleinchen smiles and murmurs her parting salute: Hals und Beinbruch ( an old German good luck greeting meaning ‘May you break your neck and legs’ ), and the great Lauritz Melchior bounds youthfully on to the Metropolitan’s aged stage.”
   
Newspapers.com
1 February 1940, Daily News (New York, NY), “Broadway” by Danton Walker, pg. 42, col. 1:
Incidental Intelligence (About Opera Stars)
Lauritz Melchior’s wife always says to him ‘Hals und Beinbruch” (I hope you break your arms and legs) when he goes onstage to sing, because to wish an opera star “Good luck” is considered the worst of all possible omens.
 
Google Books
The Scene Is Changed
By Ashley Dukes
London: Macmillan
1942
Pg. 44:
It is so important that in Middle Europe the actual wishing of luck is thought unlucky, and an actor or playwright is wished Hals-und-Beinbruch instead. This means “broken neck and legs to you”, and it serves just as well.
 
Google Books
The Theatre Handbook and Digest of Plays
Edited by Bernard Sobel
New York, NY: Crown Publishers
1948
Pg. 722:
Other superstitions include: it is bad luck to whistle in a dressing room; wearing old shoes that were associated with a hit is good luck; before a performance actors never wish each other good luck, but say “I hope you break a leg”; ...
 
25 August 1951, Morning Advocate (Baton Rouge, LA), “The Lyons Den” by Leonard Lyons, pg. 4A, col. 2:
Tradition: The next time I saw Miss Truman (Margaret Truman—ed.) on a platform was at the Runyon fund’s special performance of “Guys & Dolls,” where she served as the fund’s hostess for the evening. I, as vice-president of the fund, was to introduce her from the stage. We stood in the wings at the 46th st. theater, and when my cue came and I started to walk onstage, I heard her call to me: “Break a leg”...I wheeled, in disbelief at what I’d heard, and she repeated: “Break a leg”...She later explained this superstitution (sic) among concert artists—that it really means good luck.
   
Archive.org
August 1959, Photoplay, pg. 20,  col. 3:
“Break a leg,” says Rock (Hudson—ed.). (i.e. “good luck.”)
 
Google Books
The Playmakers
By Stuart W. Little and Arthur Cantor
New York, NY: W. W. Norton & Company, Inc.
1970
Pg. 90:
Implicit in the relationship of actor and audience are violence and hostility: the cliche phrases associated with this relationship abound in references to destruction, death, and sexual conquest.  “Merde” and “Break a leg”* are common actor’s telegrams. The victory shouts of actors after triumphant performances are filled with violent images:
   
“I killed them tonight.”
“We mowed ‘em down.”
“I laid them in the aisles.”
   
Conversely, if the show goes poorly, an actor will say, “We died out there tonight” or “We bombed” or “It just laid there.”
*Originally, Hals und Beinbruch, from the German.
 
7 April 1971, Rockford (IL) Register-Republic, “Big names cheer Marty Allen debut” by Earl Wilson, pg. 22B, col. 3:
NEW YORK—Wild-haired “Hello Dere” comedian Marty Allen arrived at the Copacabana with a handsome new boy singer named Kevin James—and got a laugh when he said he had a telegram from his former partner Steve Rossi:
 
“There’s a show business expression: ‘May you break a leg.’ May I wish you better luck than that? May you broke both legs.”
 
Playbill
32 Theatre Terms Everyone Should Know
From “break a leg” to “strike,” here’s what they mean and where they came from.
BY RUTHIE FIERBERG
AUGUST 10, 2019
(...)
Break a Leg
“It’s bad luck to say good luck on opening night,” of course, but how did we land on “break a leg”? There are a few explanations. In Ancient Greece, audiences didn’t clap at performances, they stomped. The more they stomped, the more chance there was of breaking a leg; this tradition reappeared in Elizabethan England when audiences would stomp their chairs and, again, more stomping would break the leg of the chair. Wishing someone “break a leg” is wishing for thunderous applause.