Bornholms Teater, on Østergade, is the oldest surviving theatre in any Danish provincial town, built in 1823 and still staging plays. The yellow neoclassical façade is unassuming; the interior is a perfectly preserved early-19th-century auditorium of three tiers of wooden boxes around a tiny stage, lit by candles for special performances.
You can usually get inside on the daily summer guided tours, and the schedule of plays and concerts runs through the season.