We determine the complexity of topological properties (i.e., properties closed under the Wadge equivalence) of regular ω-languages by showing that they are typically NL-complete (PSPACE-complete) for the deterministic Muller, Mostowski and Büchi automata (respectively, for the nondeterministic Rabin, Muller, Mostowski and Büchi automata). For the deterministic Rabin and Streett automata and for the nondeterministic Streett automata upper and lower complexity bounds for the topological properties are established.