Modular symmetric functions are a new class of symmetric functions which depend both on a partition ⋋ and an integer modulus p > 2. For p prime, these functions have representation theoretic significance as the irreducible characters of the general linear group G L( n, K) where K is of characteristic p. In this paper we use classical algebraic techniques to prove determinantal identities that are modular analogues to the Jacobi-Trudi, dual Jacobi-Trudi, and Giambelli identities for the classical Schur functions.