In this work, we proposed a flexible broadband metamaterial perfect absorber (FBMPA) by exploiting a pasted conductive-graphene ink on a polyimide substrate. For the flat FBMPA, an absorption over 90% was found to cover a wide frequency range (from 7.88 to 18.01 GHz). The high-absorption feature was polarization-insensitive and regarded as stable with respect to the oblique incidence up to 30 degrees of electromagnetic wave. The high absorption was maintained well even when the absorber was wrapped. That is, the FBMPA was attached to cylindrical surfaces (with the varying radius from 4 to 50 cm). For both flat and curved states, the absorption mechanism was explained by the perfect impedance matching and the dielectric loss of the proposed absorber. Our work provides the groundwork for the commercialization of future meta-devices such as sensors, optical filters/switchers, photodetectors, and energy converters.