The Palace District, or Palota Negyed, is a hip, up-and-coming area, known for its 19th-century palaces and leafy squares full of restaurants and cafes. Artsy boutique hotels, independent galleries and the Hungarian National Museum give the area a trendy, creative vibe. Cool hangouts evoking the area’s gritty past include MÜSZI, a cultural center with a bar, on the roof of a communist-era department store.