Gravity-only simulations of structure formation within the standard paradigm for the Universe are remarkably successful at reproducing observations on scales larger than about 1 Mpc. However, discrepancies begin to appear on smaller scales, indicating the importance of non-gravitational physics. The persistence of these problems to systems with ≲10^7 stars, where high dynamical mass-to-light ratios suggest dark matter physics (i.e. gravity) should dominate over baryonic physics, has long been argued for as a possible sign of new physics in the dark sector. I discuss the status of these "small-scale problems" around our galaxy, the Milky Way, with a focus on the non-gravitational physics to which they may point. I then present next-generation simulations that include realistic models for star formation and stellar feedback with enough resolution to capture low-mass dwarf galaxies in the Local Group, our region of the Universe, which yield satellite populations that broadly match the Milky Way satellites without invoking new dark matter physics. However, the simulations struggle to reproduce aspects of the satellite population of Andromeda, our galactic neighbor. Most glaringly, we lack analogues to the highest density, baryon-dominated dwarf galaxies around Andromeda. Moving forward, new instruments will yield even tighter constraints by uncovering new (fainter) galaxies and probing existing objects to even greater depth. Reproducing the full diversity of galactic structure in the nearby Universe, particularly given these new data, will continue to be a stringent test of galaxy formation models.