This talk will discuss novel critical behavior and ``Griffiths-McCoy" singularities (which occur away from criticality) near quantum phase transitions in disordered systems with discrete (Ising) symmetry. One finds that distributions of many quantities are very broad, so it is insufficient to simply take the average. After reviewing some exact results in one-dimension obtained using a real space renormalization group approach, I will discuss extensions of these results obtained by a numerical method in which the spin problem is mapped on to free fermions. In this way, whole distributions can be determined with precision. The talk will then describe quantum Monte Carlo simulations which show that similar results also occur in higher dimensions, at least for $d=2$.
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