The gapless surface states of three-dimensional topological insulators are a new form of matter, and there is much active research on exotic order on the surface of topological insulators due to electron-electron interactions. In this talk, we investigate electron-electron interactions on the surface of a three-dimensional topological insulator. First, we construct a phenomenological Landau theory for the two-dimensional helical Fermi liquid found on the surface of a three-dimensional time-reversal invariant topological insulator. By projecting quasiparticle states onto the Fermi surface, we obtain an effectively spinless, Landau theory with a single Landau parameter per angular momentum channel that captures the spin-momentum locking or nontrivial Berry phase of the Fermi surface. Next, we derive equilibrium properties, criteria for Fermi surface instabilities, and collective mode dispersions in terms of the projected Landau parameters. In particular, we investigate the nematic instability of the helical Fermi surface in detail and discuss various observable consequences of nematic order unique to helical Fermi liquids.