Osh and Alamedin bazaars are both popular for food and handicrafts.
The Art Gallery sells paintings and traditional Kyrgyz products (particularly popular are embroidered Kyrgyz felt hats [kalpak], felt carpets and chess sets with traditional Kyrgyz figures).
Some souvenir shopping also can be done in TSUM (city center just across the main post office). TSUM is the biggest supermarket in Bishkek and offers everything from toiletries to new electronics. Souvenirs are located on the fourth level (some negotiation is expected).
Beta Stores is on Chui Ave / Isanova St. intersection and Europa supermarket (Pravda St. / Toktogula St., 5 minutes walk from TSUM) also have souvenirs but they can be overpriced.