Lattice QCD using fermions whose Dirac operator obeys the Ginsparg-Wilson relation is perhaps the best known formulation of QCD with a finite cutoff. It reproduces all the low energy QCD phenomenology associated with chiral symmetry at finite lattice spacings. In particular it explains the origin of massless pions due to spontaneous chiral symmetry breaking and leads to new ways to approach the U(1) problem on the lattice. Here we show these results in the path integral formulation and derive for the first time in lattice QCD a known formal continuum relation between the chiral condensate and the topological susceptibility. This relation leads to predictions for the critical behavior of the topological susceptibility near the phase transition and can now be checked in Monte Carlo simulations even at finite lattice spacings. ©1999 The American Physical Society.