We consider certain decision problems about PL, locally flat, closed, connected, orientable surfaces in S^4. In particular we determine explicit integers g_1 and g_2 such that if g is at least g_1 (resp. g_2) then there is no algorithm to decide whether or not a surface in S^4 of genus g is PL (resp. TOP) unknotted. The proofs depend on results about surface groups, i.e. the fundamental groups of complements of surfaces in S^4. For example we determine explicit integers g_3 and g_4 such that there is a surface in S^4 of genus g_3 whose group has unsolvable word problem, and a surface in S^4 of genus g_4 whose group contains an isomorphic copy of every finitely presented group.