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Point-scanning two-photon microscopy enables high-resolution imaging within scattering specimens such as the mammalian brain, but sequential acquisition of voxels fundamentally limits imaging speed. We developed a two-photon imaging technique that scans lines of excitation across a focal plane at multiple angles and uses prior information to recover high-resolution images at over 1.4 billion voxels per second. Using a structural image as a prior for recording neural activity, we imaged visually-evoked and spontaneous glutamate release across hundreds of dendritic spines in mice at depths over 250 microns and frame-rates over 1 kHz. Dendritic glutamate transients in anaesthetized mice are synchronized within spatially-contiguous domains spanning tens of microns at frequencies ranging from 1-100 Hz. We demonstrate high-speed recording of acetylcholine and calcium sensors, 3D single-particle tracking, and imaging in densely-labeled cortex. Our method surpasses limits on the speed of raster-scanned imaging imposed by fluorescence lifetime.