Mubarakiya, a market stretching back 200 years,  is certainly the best known and most often visited historical/heritage site in Kuwait. Bazaar has a helpful map here. The artist Zahra Marwan made a beautiful pictorial map of Mubarakiya a few years ago, as part of a project with Huda Abdulmughni, which you can see above. Here are some of the sites within Mubarakiya:

• Safat Square
• The popular restaurant Shater Abbas and the oldest restaurant in Kuwait
• The Kiskh or Kiosk of Sheikh Mubarak
• Historic Water Market
• Souq Al Manakh
• Gulf Bank Headquarters
• Historic Post Office
• Mubarakiya School
• The National Library Bookstore
• Souk Al-Kabir Mosque