Known to most locals as Centru Vechi (the Old Centre), Bucharest's Old Town is defined by the area bordered by the Dambovita river to the south, Calea Victoriei to the west, Bulevardul Brătianu to the east and Regina Elisabeta to the north. The area is more or less all that’s left of pre-World War II Bucharest. What the war didn’t destroy (and it destroyed a fair bit: allied bombing was fierce during the early part of 1944) communism did, most notably in the form of the grandiose Civic Centre project that saw almost a fifth of the total area of the city flattened to make way for Bulevardul Unirii and Casa Poporului. Indeed, that anything survives at all is little short of a miracle.
All of that changed however in the wake of the fire at Colectiv in October of last year, and a number of venues which were housed in buildings considered an earthquake risk (those with a red disc on them) have been (quite properly) forced to close by new legislation.
As such, the future of Old Town is a little unsure. What you can be sure of is that those venues which remain open in the Old Town (less a Red Light and more a Red Disc district these days) are as safe and secure as Bucharest gets.