High-mobility two-dimensional (2D) semiconductors are desirable for high-performance mechanically flexible nanoelectronics. In this work, we report the first flexible black phosphorus (BP) field-effect transistors (FETs) with electron and hole mobilities superior to what has been previously achieved with other more studied flexible layered semiconducting transistors such as MoS2 and WSe2. Encapsulated bottom-gated BP ambipolar FETs on flexible polyimide afforded maximum carrier mobility of about 310 cm2/V·s with field-effect current modulation exceeding 3 orders of magnitude. The device ambipolar functionality and high-mobility were employed to realize essential circuits of electronic systems for flexible technology including ambipolar digital inverter, frequency doubler, and analog amplifiers featuring voltage gain higher than other reported layered semiconductor flexible amplifiers. In addition, we demonstrate the first flexible BP amplitude-modulated (AM) demodulator, an active stage useful for radio receivers, based on a single ambipolar BP transistor, which results in audible signals when connected to a loudspeaker or earphone. Moreover, the BP transistors feature mechanical robustness up to 2% uniaxial tensile strain and up to 5000 bending cycles.