Asmara's strong point are its buildings, rather than its museums, but it's also worth popping your head into the National Museum, west of the former Ghidei (Governor's) Palace. Cameras and cell phones must be left with the front-desk attendant.
The exhibition space, about half of which is currently utilised, is dedicated to a collection of archaeological finds, taxidermy, medieval artefacts, ethnography exhibits and national artwork. The latter two will be of greatest interest to most travellers, showcasing a collection of household objects from some of the country's nine major ethnic groups, alongside paintings that explore Eritrean history from the tribal and colonial periods through to the war and independence.