Starting from the random triangulation definition of two-dimensional euclidean quantum gravity, we define the continuum limit and compute the partition function for closed surfaces of any genus. We discuss the appropriate way to define continuum string perturbation theory in these systems and show that the coefficients (as well as the critical exponents) are universal. The universality classes are just the multicritical points described by Kazakov. We show how the exact non-perturbative string theory is described by a non-linear ordinary differential equation whose properties we study. The behavior of the simplest theory, c = 0 pure gravity, is governed by the Painlevé transcendent of the first kind.
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