We introduce wave encoded acquisition and reconstruction techniques for highly accelerated echo planar imaging (EPI) with reduced g-factor penalty and image artifacts. Wave-EPI involves playing sinusoidal gradients during the EPI readout while employing interslice shifts as in blipped-CAIPI acquisitions. This spreads the aliasing in all spatial directions, thereby taking better advantage of 3D coil sensitivity profiles. The amount of voxel spreading that can be achieved by the wave gradients during the short EPI readout period is constrained by the slew rate of the gradient coils and peripheral nerve stimulation (PNS) monitor. We propose to use a half-cycle sinusoidal gradient to increase the amount of voxel spreading that can be achieved while respecting the slew and stimulation constraints. Extending wave-EPI to multi-shot acquisition minimizes geometric distortion and voxel blurring at high in-plane resolution, while structured low-rank regularization mitigates shot-to-shot phase variations without additional navigators. We propose to use different point spread functions (PSFs) for the k-space lines with positive and negative polarities, which are calibrated with a FLEET-based reference scan and allow for addressing gradient imperfections. Wave-EPI provided whole-brain single-shot gradient echo (GE) and multi-shot spin echo (SE) EPI acquisitions at high acceleration factors and was combined with g-Slider slab encoding to boost the SNR level in 1mm isotropic diffusion imaging. Relative to blipped-CAIPI, wave-EPI reduced average and maximum g-factors by up to 1.21- and 1.37-fold, respectively. In conclusion, wave-EPI allows highly accelerated single- and multi-shot EPI with reduced g-factor and artifacts and may facilitate clinical and neuroscientific applications of EPI by improving the spatial and temporal resolution in functional and diffusion imaging.