A pushdown automaton is said to make a turn at a given instant if it changes at that instant from stack increasing to stack decreasing. Let \hboxNPDA-TURN(\,f(n)) and \hboxDPDA-TURN(\,f(n)) denote the classes of languages accepted by nondeterministic and deterministic pushdown automata respectively that make at most f ( n ) turns for any input of length n . In this paper the following inclusions that express the space complexity of turn bounded pushdown automata are given: \hboxDPDA-TURN(\,f(n)) \subseteq \bf DSPACE(\log f(n)\log n) , and \hboxNPDA-TURN(\,f(n)) \subseteq \bf NSPACE (\log f(n)\log n) . In particular, it follows that finite-turn pushdown automata are logarithmic space bounded: \hboxDPDA-TURN(O(1))\subseteq \bf DL and \hboxNPDA-TURN(O(1))\subseteq \bf NL , from which two corollaries follow: one is that the class of metalinear context-free languages is complete for NL , and the other is that a more tight inclusion \hboxNPDA-TURN(\,f(n)) \subseteq \bf DSPACE(\log^2 f(n)\log n) cannot be derived unless \bf DL=\bf NL , though \hboxNPDA-TURN (\,f(n)) \subseteq \bf DSPACE(\log^2 f(n)\log^2n) holds.