Resilient functions play an important role in the art of information security. In this correspondence, we discuss the existence, construction, and enumeration of resilient functions over finite fields. We show that, for each finite field GF(q) with q > 3, we can easily construct a large number of (q, n, 1, n - 1) resilient functions, most of which include mixing terms. We give a general structure for (q, m + 1, m, 1) resilient functions, and present an example which is not of this general structure. We prove that (q, m + 2, m, 2) resilient functions exist for any m such that 1 2. We prove that (q, m + t, m, t) resilient functions exist for any (m, t) such that 1 3. By making some simple generalizations of former results, we also provide some new methods for constructing resilient functions.