Let L be an algebraic function field in k parameters t1,..,tk. Let f1, f2 be non-zero polynomials in L[x]. We give two algorithms for computing their gcd. The first, a modular GCD algorithm, is an extension of the modular GCD algorithm of Brown for Z[x1,..,xn] and Encarnacion for Q(alpha)[x] to function fields. The second, a fraction-free algorithm, is a modification of the Moreno-Maza and Rioboo algorithm for computing gcds over triangular sets. The modification reduces coefficient growth in L to be linear. We give an empirical comparison of the two algorithms using implementations in Maple.