While modern models of galaxy evolution have made significant strides in explaining the observed properties of massive galaxies over cosmic time, recent observational results have illustrated a fundamental problem with the ability of these same models to predict the evolution of low-mass systems. In particular, simulations of galaxy evolution tend to overpredict the number of passive (or "quenched") low-mass galaxies at low and intermediate redshift. Using data from the SDSS, I will present recent work to constrain the quenching of satellite galaxies in the local Universe, with a focus on identifying the host and satellite properties critical to the cessation of star formation at low stellar masses. This analysis, when combined with observations of the Local Group, points towards a characteristic mass scale for satellite quenching as well as strong limits to our understanding of dwarf galaxies. As time permits, I will present ongoing efforts to expand our knowledge of this low-mass galaxy frontier.