We describe an automata-theoretic approach to the automated checking of truth and validity for temporal logics. The basic idea underlying this approach is that for any formula we can construct an alternating automaton that accepts precisely the models of the formula. For linear temporal logics the automaton runs on infinite words while for branching temporal logics the automaton runs on infinite trees. The simple combinatorial structures that emerge from the automata-theoretic approach decouple the logical and algorithmic components of truth and validity checking and yield clean and essentially optimal algorithms for both problems.