Wide applications of personal consumer electronics have triggered tremendous need for portable power sources featuring light-weight and mechanical flexibility. Perovskite solar cells offer a compelling combination of low-cost and high device performance. Here we demonstrate high-performance planar heterojunction perovskite solar cells constructed on highly flexible and ultrathin silver-mesh/conducting polymer substrates. The device performance is comparable to that of their counterparts on rigid glass/indium tin oxide substrates, reaching a power conversion efficiency of 14.0%, while the specific power (the ratio of power to device weight) reaches 1.96 kW kg(-1), given the fact that the device is constructed on a 57-μm-thick polyethylene terephthalate based substrate. The flexible device also demonstrates excellent robustness against mechanical deformation, retaining >95% of its original efficiency after 5,000 times fully bending. Our results confirmed that perovskite thin films are fully compatible with our flexible substrates, and are thus promising for future applications in flexible and bendable solar cells.