We introduce acquisition and reconstruction strategies for robust, high-quality multi-shot EPI (msEPI) without phase navigators. We extend the MUSSELS low-rank constrained parallel imaging technique to perform Virtual Coil (VC) reconstruction, and demonstrate diffusion imaging with sub-millimeter in-plane resolution using 55% partial-Fourier (PF) sampling. We propose Blip Up-Down Acquisition (BUDA) using interleaved blip-up and -down phase encoding, and incorporate B0 forward-modeling into MUSSELS to enable distortion- and navigator-free msEPI. We improve the acquisition efficiency by developing Simultaneous MultiSlice (SMS-)MUSSELS, and combine it with machine learning (ML) to provide Rtotal=16-fold acceleration with 3-shots. Deploying this in a spin-and-gradient-echo (SAGE) scan with signal modeling allows for whole-brain T2 and T2* mapping with high geometric fidelity in 12.5 seconds.