In this talk, I review the basic physical picture of galaxy formation offered by empirical models connecting galaxies to the dark matter halos they live in. I show that many of the well-established trends in galaxy evolution can be understood through quite simple models of the galaxy-halo connection. I will give examples of how these simple models can be employed to constrain cosmological parameters, as well as how they inform more complex semi-analytical models of galaxy formation and state-of-the-art hydrodynamical simulations. However, recent observational and theoretical developments have revealed that the traditional approach to the galaxy-halo connection is too simplistic to reliably model observed trends in star-formation and quenching at low redshift. This inadequacy presents a new challenge both to the theory of large-scale structure, and to the precision cosmology program. I conclude by outlining a plan for the path towards a new generation of galaxy-halo models, so that we may reap the benefits of deep and wide-field galaxy surveys to robustly constrain cosmological parameters with measurements of large-scale structure, and to develop a more complete picture of galaxy evolution.