I am presuming this question refers to the American NATIONAL THEATRE or what was once called Ford's theatre, where President Lincoln was shot. There was, and may still be a National Theatre on Broadway, but that is a Broadway theatre just as any other, and is not the one in Washington, DC.
The National Theatre is known to be "America's First Theatre", and has gone through many changes over time. It burned down, and a new theatre was built. It hoasted the openings of many major performances and either the greatest artists on tour, or occasionally parallel companies to ones running on Broadway. This is the theatre where Presidents' have been entertained, along with dignitaries, and occasionally, been roasted!
But there is a very dark part of our National Theatre History too. It did not allow persons of color to be admitted, until after a protest while President Truman was watching the 25th anniversary edition of Blossom Time, which starred among others, Joseph Toner. He as an activist with Actor's Equity Association, joined in a strike of the theatre, which went to the Supreme Court. A Supreme court decision was handed down ordering that the theatre be integrated, and that was unacceptable to the director of the theatre, which essentially meant the Theatre Guild at that time. Those with the most power were Rodgers and Hammerstein II, and naturally the Shubert Org. which then ran what was termed "Shubert's Alley", given their monopoly on the Theatre industry. The federal government did not enforce the Supreme Court's order, and for several years, it was turned into a movie house.
Other issues that are problematic with our National Theatre, is that the persons who were accepted to put a database together were none other than the Shubert Org. I have downloaded documents that show imbedded information, and this has not been translated into real documentation of real people. Thus the use of pseudonyms for many theatre artists continues. In other ways, the Databse, even with these flaws, can be very interesting. For example, a person named William Kent is noted as the star of the touring 1932 Show Boat after it's run, with Jules Bledose, which happens to fit into other documentation. For researcher the source was invaluable to seek information on whether what we think happened on Broadway was so.
I leave you with this quote from a National Theatre Playbill dated Jan 16, 1961. "The National Theatre was built in 1835 on its present location, and is the oldest stage theatre in the United States that during its entire history has been used only for the entertaining arts. Junius Brutus Booth, Emma Wheatley and Madame Celeste were stars of its first season. In their book, "A Great Curtain Falls," Atkinslon and Kiraly write that the National Theatre is revered by "all American citizens because it is the link to their dramatic past. There is no great actor of the modern era whose name is not inseparably connected with the National....it is historic ground, sacred to Thalia, Terpsucore and Melpomene."
Perhaps romanticized, but it was the site of the first civil rights action, and was our artists that did so, even though some lost their careers for it, including Joseph Toner. Perhaps it is sacred ground, after all.
The National Theatre is a producing house.
National Theatre Bucharest was created in 1852.
Montenegrin National Theatre was created in 1953.
New National Theatre Tokyo was created in 1997.
Cluj-Napoca National Theatre was created in 1906.
The National Theatre is a producing house.
National Theatre of Scotland was created in 2006.
Ethiopian National Theatre was created in 1955.
Iaşi National Theatre was created in 1896.
Serbian National Theatre was created in 1861.
Palestinian National Theatre was created in 1984.
Royal National Theatre was created in 1976.
Sarajevo National Theatre was created in 1921.
National Comedy Theatre was created in 2000.
National Theatre - Oslo - was created in 1899.
Latvian National Theatre was created in 1919.
Finnish National Theatre was created in 1902.