We review our recent and ongoing work with Fermi gases on an atom chip. After reviewing some statistical and thermodynamic properties of the ideal, non-interacting Fermi gas, and a brief description of our atom chip and its capabilities, we discuss our experimental approach to producing a potassium-40 degenerate Fermi gas (DFG) using sympathetic cooling by a rubidium-87 Bose-Einstein condensate on an atom chip. In doing so, we describe the factors affecting the loading efficiency of the atom chip microtrap. This is followed by a discussion of species selectivity in radio frequency manipulation of the Bose-Fermi mixture, which we explore in the context of sympathetic evaporative cooling and radio-frequency dressed adiabatic double-well potentials. Next, we describe the incorporation of a crossed-beam dipole trap into the atom chip setup, in which we generate and manipulate strongly interacting spin mixtures of potassium-40. Finally, we conclude with a brief discussion of future research directions with DFGs and atom chips.