A recently introduced model-based deep learning (MoDL) technique successfully incorporates convolutional neural network (CNN)-based regularizers into physics-based parallel imaging reconstruction using a small number of network parameters. Wave-controlled aliasing in parallel imaging (CAIPI) is an emerging parallel imaging method that accelerates imaging acquisition by employing sinusoidal gradients in the phase- and slice/partition-encoding directions during the readout to take better advantage of 3D coil sensitivity profiles. We propose wave-encoded MoDL (wave-MoDL) combining the wave-encoding strategy with unrolled network constraints for highly accelerated 3D imaging while enforcing data consistency. We extend wave-MoDL to reconstruct multicontrast data with CAIPI sampling patterns to leverage similarity between multiple images to improve the reconstruction quality. We further exploit this to enable rapid quantitative imaging using an interleaved look-locker acquisition sequence with T₂ preparation pulse (3D-QALAS). Wave-MoDL enables a 40 s MPRAGE acquisition at 1 mm resolution at 16-fold acceleration. For quantitative imaging, wave-MoDL permits a 1:50 min acquisition for T₁, T₂, and proton density mapping at 1 mm resolution at 12-fold acceleration, from which contrast-weighted images can be synthesized as well. In conclusion, wave-MoDL allows rapid MR acquisition and high-fidelity image reconstruction and may facilitate clinical and neuroscientific applications by incorporating unrolled neural networks into wave-CAIPI reconstruction.