High-speed actuation of laser frequency(1)is critical in applications using lasers and frequency combs(2,3), and is a prerequisite for phase locking, frequency stabilization and stability transfer among optical carriers. For example, high-bandwidth feedback control of frequency combs is used in optical-frequency synthesis(4), frequency division(5)and optical clocks(6). Soliton microcombs(7,8)have emerged as chip-scale frequency comb sources, and have been used in system-level demonstrations(9,10). Yet integrated microcombs using thermal heaters have limited actuation bandwidths(11,12)of up to 10 kilohertz. Consequently, megahertz-bandwidth actuation and locking of microcombs have only been achieved with off-chip bulk component modulators. Here we demonstrate high-speed soliton microcomb actuation using integrated piezoelectric components(13). By monolithically integrating AlN actuators(14)on ultralow-loss Si(3)N(4)photonic circuits(15), we demonstrate voltage-controlled soliton initiation, tuning and stabilization with megahertz bandwidth. The AlN actuators use 300 nanowatts of power and feature bidirectional tuning, high linearity and low hysteresis. They exhibit a flat actuation response up to 1 megahertz-substantially exceeding bulk piezo tuning bandwidth-that is extendable to higher frequencies by overcoming coupling to acoustic contour modes of the chip. Via synchronous tuning of the laser and the microresonator, we exploit this ability to frequency-shift the optical comb spectrum (that is, to change the comb's carrier-envelope offset frequency) and make excursions beyond the soliton existence range. This enables a massively parallel frequency-modulated engine(16,17)for lidar (light detection and ranging), with increased frequency excursion, lower power and elimination of channel distortions resulting from the soliton Raman self-frequency shift. Moreover, by modulating at a rate matching the frequency of high-overtone bulk acoustic resonances(18), resonant build-up of bulk acoustic energy allows a 14-fold reduction of the required driving voltage, making it compatible with CMOS (complementary metal-oxide-semiconductor) electronics. Our approach endows soliton microcombs with integrated, ultralow-power and fast actuation, expanding the repertoire of technological applications of microcombs. By monolithically integrating piezoelectric actuators on ultralow-loss photonic circuits, soliton microcombs-a spectrum of sharp lines over a range of optical frequencies-can be modulated at high speeds with megahertz bandwidths.