  We study zero temperature phase transitions in two classes of random quantum
systems -the $q$-state quantum Potts and clock models. For models with purely
ferromagnetic interactions in one dimension, we show that for strong randomness
there is a second order transition with critical properties that can be
determined exactly by use of an RG procedure. Somewhat surprisingly, the
critical behaviour is completely independent of $q$ (for $2 \leq q < \infty$).
For the $q > 4$ clock model, we suggest the existence of a novel multicritical
point at intermediate randomness. We also consider the $T = 0$ transition from
a paramagnet to a spin glass in an infinite range model. Assuming that the
transition is second order, we solve for the critical behaviour and find $q$
independent exponents.
