Flexible thin film solar cells such as CIGS, CdTe, and a-Si:H have received worldwide attention. Until now, Si solar cells dominate the photovoltaic market. Its production cost is a major concern since Si substrates account for the major cost. One way to reduce the module production cost is to use the low-cost flexible substrates. It reduces the installation and transportation charges also, thereby reducing the system price. Apart from metallic foils, plastic films and flexible glass, paper substrates such as cellulose papers, bank notes, security papers and plain white copying papers are also used as substrates for flexible solar cells. In this review, recent developments in flexible CIGS, CdTe and a-Si:H solar cells are reported. Progress on various flexible foils, fabrication and stability issues, current challenges and solutions to those challenges of using flexible foils, and industrial scenario are reviewed in detail. Encapsulation issues and solutions related to water vapor transmission rate are discussed.