Dwarf galaxies are among the most numerous objects in the Universe, and also appear to be the most dark matter (DM) dominated; consequently, they provide strong tests on the standard paradigm of hierarchical galaxy formation: cold dark matter with a cosmological constant (ΛCDM). Due to their low luminosities, however, observational studies of dwarfs have remained limited to the nearby Universe, with a primary focus on the satellites of the Milky Way (MW). Upcoming surveys will relax the observational constraints, allowing for studies of dwarf galaxies well beyond the virial radius (Rv) of the MW, where the presence of the Andromeda (M31) galaxy may have a measurable impact. In this thesis, I introduce the ELVIS (Exploring the Local Volume in Simulations) suite, which resolves the formation and evolution of Local Group (LG)-like pairs of galaxies, chosen to resemble the MW and M31 host halos, along with thousands of dwarf halos that surround them. Using ELVIS, I demonstrate that simulations of isolated MW-size hosts do not correctly predict dwarf counts and kinematics beyond Rv of the MW, and also explore the faint-end relationship between stellar mass (Mstar) and halo mass (Mhalo). I then use ELVIS to explore the “too-big- to-fail” (TBTF) problem, a challenge for ΛCDM that points out the overabundance of large (maximum circular velocity Vmax ~ 30 km/s) dwarf halos relative to observations, finding TBTF to be ubiquitous to MW-size hosts and the fields around them. Finally, I explore two possible explanations for TBTF in depth: fluctuations in the baryonic mass in the center of
a dwarf halo, driven by supernovae feedback, which I find lacks the energy to solve TBTF, and a “running” initial power spectrum, as motivated by the BICEP-2 measurement of primordial gravitational waves, which significantly alleviates the problem and may enhance the impact of other processes, including baryonic feedback.