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We describe new results in parametrized complexity theory. In particular, we prove a number of concrete hardness results for W[P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parametrized complexity of analogues of PSPACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W [P] and related classes. For instance, we show that W[P] can be characterized in terms of the DTIME (2^o(n)) and NP.