  We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture
for half-integral weight automorphic forms. Assuming the Generalized Riemann
Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic
Hecke cusp forms for $\Gamma_0(4)$ lying in Kohnen's plus subspace and for
half-integral weight Hecke Maa{\ss} cusp forms for $\Gamma_0(4)$ lying in
Kohnen's plus subspace. By combining the former result along with an argument
of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp
equidistribute with respect to hyperbolic measure on $\Gamma_0(4)\backslash
\mathbb H$ as the weight tends to infinity.
