In optical microscopy, the slow axial scanning rate of the objective or the sample has traditionally limited the speed of volumetric imaging. Recently, by conjugating either a movable mirror to the image plane in a remote-focusing geometry or an electrically tuneable lens (ETL) to the back focal plane, rapid axial scanning has been achieved. However, mechanical actuation of a mirror limits the axial scanning rate (usually only 10-100 Hz for piezoelectric or voice coil-based actuators), while ETLs introduce spherical and higher-order aberrations that prevent high-resolution imaging. In an effort to overcome these limitations, we introduce a novel optical design that transforms a lateral-scan motion into a spherical aberration-free axial scan that can be used for high-resolution imaging. Using a galvanometric mirror, we scan a laser beam laterally in a remote-focusing arm, which is then back-reflected from different heights of a mirror in the image space. We characterize the optical performance of this remote-focusing technique and use it to accelerate axially swept light-sheet microscopy by an order of magnitude, allowing the quantification of rapid vesicular dynamics in three dimensions. We also demonstrate resonant remote focusing at 12 kHz with a two-photon raster-scanning microscope, which allows rapid imaging of brain tissues and zebrafish cardiac dynamics with diffraction-limited resolution.