They bear such famous names, known throughout the world, that to the outsider and the tourist it seems unthinkable that they should sparkle with anything less than success.
One of them, the National Theatre, stands on the south bank of the Thames, alive with three plays a night, its glass-walled lobbies awash with up to 2,000 playgoers and the sounds of eating, drinking, and entertainment including pianists, clarinet quartets, minstrels, and a Celtic harp or two.
The other is deep inside London's newest cultural center, - the Barbican - in London's financial district. The butt of many a joke because it is both hard to find and to navigate within, the cavernous concrete building is nonetheless prestigious and popular.
The Royal Shakespeare Company (RSC) has benefited by moving in - and it maintains its reputation as perhaps the most famous theater in the English-speaking world in its other headquarters beside the peaceful Avon River in Stratford-on-Avon to the north.
Yet for all their excellence and reputation, and despite government subsidies of millions of pounds a year, both the National and the RSC face complicated financial futures. And both are competing - discreetly, courteously, but determinedly - for more funds.
Britain remains mired in economic recession, despite more optimistic noises now being heard that lower oil prices and lower interest rates are beginning to stimulate a recovery. The result is that each company is lobbying for extra funds. Each wants more government money, and each has turned to discounts, promotion, and television productions of its plays to make ends meet.
* The RSC claims to feel the pinch most sharply. It runs four separate theaters - two in Stratford-on-Avon (the Shakespeare Theatre and the much smaller one called the Other Place) and two in the Barbican in London (the main theater and a smaller one called the Pit).
Talk to David Brierley, general manager of the RSC, and he says that in Britain government subsidies are small compared to Western Europe and cannot banish financial woes.
''For 1983-84,'' Mr. Brierley says briskly, ''the government granted us (STR) 3.6 million ($5.4 million). That was only about 40 percent of its operating costs, and (STR)50,000 below the year before. Stratford itself is only a small place, with about 100,000 people in its catchment area, and gives us no financial aid at all.''
What distresses him and RSC director Trevor Nunn is not only what they see as the smallness of the grant, but also the fact that the rival National Theatre gets so much more.
''The National,'' he states, ''has about the same size of company, between 120 and 130 actors, but one fewer auditorium to fill. In 1980-81 it played to 64 ,000 fewer people.''
According to the National's general administrator, Michael Elliott, the National received (STR)6.39 million in 1983-84. In addition, it expected the Greater London Council, the largest local government authority in Western Europe , to continue its previous year's grant of (STR)690,000.
''What that means,'' says Mr. Brierley, ''is that every single seat at the National was subsidized last year to the amount of nine pounds and 46 pence ($14 .19), whereas RSC tickets were subsidized by only three pounds and 62 pence ($5. 43). Our ticket prices have to be higher. That's a terrible discrimination against the RSC which we feel is very wrong.''
So anxious are Mr. Brierley and Mr. Nunn to make their point that Mr. Brierley adds, ''the discrimination is making it almost impossible for the RSC to carry on.''
In February last year, a 35-page memorandum to the RSC governors said reserves built up in earlier years were exhausted. In April, the governors argued how best to save money, and suggestions were made that one of the major RSC theaters would have to close. In October Mr. Nunn said it was unthinkable to do any such thing.
The RSC does not argue that the National gets too much. Quite the reverse: ''By any international standards the National is dangerously underfunded,'' Mr. Brierley says.
The RSC is lobbying the House of Commons, the Arts Council, and the minister for the arts, Paul Channon, for more funds. It has markedly stepped up promotion campaigns. It is encouraged at having received an extra subsidy from the Arts Council last year. This left (STR)200,000 over to reduce accumulated losses which amounted to (STR)385,000 a year ago.
And the Corporation of the City of London gives a hidden subsidy to the RSC by charging it a less-than-commercial rent for the Barbican Theatre.
Meanwhile, Brierley and his staff work hard to generate other income. The RSC production of ''Nicholas Nickleby'' was a hit on Broadway, and the RSC has joined forces with the British Broadcasting Corporation and other television companies since 1962. The RSC has just signed a new contract with RKO General Inc. in the United States for regular televised plays for American audiences.
Contracts of this sort, and the increase in audience sizes after a drop in the late 1970s, sound promising. But sharp price increases run the risk of turning people away again.
At the National Theatre, Michael Elliott also works overtime to generate funds.
The three National auditoriums - the Olivier, the Lyttleton, and the Cottesloe Theatres - put on about 12 productions each month in repertory, filling on average 80 percent of the total of 2,400 seats per night. Mr. Elliott and director Sir Peter Hall note proudly that audiences were up 8 percent last year. So far in 1983 they are 11 percent higher.
The company spends $:11 to $:12 million a year. About $:2.5 million is needed to keep the 4 1/2-acre building open before a single play is staged, Mr. Elliott says. Energy costs are particularly high, and every effort is made to cut down backstage.
(The RSC, which does not own one of its theaters, the Barbican, says its own running costs are $:900,000 including upkeep of half a mile of Avon River bank.)
Director Sir Peter Hall goes over last night's audience figures when he arrives at his offices every morning. Does he, then, feel pressure to ''go commercial,'' to lower the National's sights from time to time, to keep seats filled?
''No,'' he says. ''It's a question of balance. Our subsidies allow us to try other things - 'Oresteia' by Aeschylus, for instance, and 'Schweyk in the Second World War.'
''Sir Peter agrees that if he put on translations from the Danish every night , seats would empty. He does try for one seat-filler each night - ''Guys and Dolls,'' for instance, and ''The Importance of Being Earnest'' with Judi Dench as Lady Bracknell.
The National generally receives good reviews as a company. Robert Cushman, theater critic for the national newspaper, The Observer, said in an interview: ''There's nothing wrong with the National having full houses, nothing wrong in doing ''Guys and Dolls,'' which is a classic. . . . I think the National is a success, up to a point. . . . I find it quite hard to keep track of its identity as a company, and its plays seem to come and go rather fast.''
Sir Peter regrets several shortcomings: not enough being done for children; no regional British companies coming to the premises while the National goes on tour; no foreign companies coming here, which are ''too expensive and we can't get anyone here to help pay the costs.''
Meanwhile, the company helps keep alive the craft of the theater, from acting to making every conceivable prop. Except shoes - ''cheaper to buy them,'' Mr. Elliott says.