Incidence bounds for non-linear objects in finite fields have been elusive, with results being known only for specific types of curves. In this talk we give a brief overview of this topic, and outline the proofs of two new non-linear incidence results, the first being joint work with James Wheeler, and the second being joint work with Thang Pham and Ali Mohammadi. These results pertain to Mobius transforms and general sets of conics, respectively.