The canonical quantization of supergravity is developed, starting from the Hamiltonian treatment of classical supergravity. Quantum states may be represented by wave functionals $f({{e}^{A{A}^{\ensuremath{'}}}}_{i}(x),{{\ensuremath{\psi}}^{A}}_{i}(x))$ of the spatial spinor-valued tetrad forms ${{e}^{A{A}^{\ensuremath{'}}}}_{i}$ and of the right-handed spatial part ${{\ensuremath{\psi}}^{A}}_{i}$ of the spin-$\frac{3}{2}$ field, or equivalently by functionals $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{f}({{e}^{A{A}^{\ensuremath{'}}}}_{i}(x),{{\overline{\ensuremath{\psi}}}^{{A}^{\ensuremath{'}}}}_{i}(x))$ of ${{e}^{A{A}^{\ensuremath{'}}}}_{i}$, and the left-handed part ${{\overline{\ensuremath{\psi}}}^{{A}^{\ensuremath{'}}}}_{i}$. In the first representation the momentum ${{p}_{A{A}^{\ensuremath{'}}}}^{i}$ classically conjugate to ${{e}^{A{A}^{\ensuremath{'}}}}_{i}$, together with ${{\overline{\ensuremath{\psi}}}^{{A}^{\ensuremath{'}}}}_{i}$, can be represented by functional differential operators such that the correct (anti) commutation relations hold; similarly for ${{p}_{A{A}^{\ensuremath{'}}}}^{i}$,${{\ensuremath{\psi}}^{A}}_{i}$ in the second representation. A formal inner product can be found in which ${{p}_{A{A}^{\ensuremath{'}}}}^{i}$ is Hermitian and ${{\ensuremath{\psi}}^{A}}_{i}$,${{\overline{\ensuremath{\psi}}}^{{A}^{\ensuremath{'}}}}_{i}$ are Hermitian adjoints. Physical states obey the quantum constraints ${J}_{\mathrm{AB}}f=0$, ${\overline{J}}_{{A}^{\ensuremath{'}}{B}^{\ensuremath{'}}}f=0$, ${S}_{A}f=0$, ${\overline{S}}_{{A}^{\ensuremath{'}}}f=0$, ${\mathcal{H}}_{A{A}^{\ensuremath{'}}}f=0$, where ${J}_{\mathrm{AB}}$, ${\overline{J}}_{{A}^{\ensuremath{'}}{B}^{\ensuremath{'}}}$ are the quantum versions of the classical generators of local Lorentz rotations, ${S}_{A}$, ${\overline{S}}_{{A}^{\ensuremath{'}}}$ correspond to classical supersymmetry generators, and ${\mathcal{H}}_{A{A}^{\ensuremath{'}}}$ to generalized coordinate transformations. The constraints ${J}_{\mathrm{AB}}f=0$, ${\overline{J}}_{{A}^{\ensuremath{'}}{B}^{\ensuremath{'}}}f=0$ describe the invariance of $f(e,\ensuremath{\psi})$ under local Lorentz transformations, ${\overline{S}}_{{A}^{\ensuremath{'}}}f=0$ gives a simple transformation property of $f(e,\ensuremath{\psi})$ under left-handed supersymmetry transformations applied to ${{e}^{A{A}^{\ensuremath{'}}}}_{i}$, ${{\ensuremath{\psi}}^{A}}_{i}$, and ${S}_{A}f=0$ gives a corresponding property of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{f}(e,\overline{\ensuremath{\psi}})$ under right-handed transformations; these transformation properties are all that is required of a physical state. All physical wave functionals can be found by superposition from the amplitude $K$ to go from prescribed data ${({{e}^{A{A}^{\ensuremath{'}}}}_{i},{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}}^{{A}^{\ensuremath{'}}}}_{i})}_{I}$ on an initial surface to data ${({{e}^{A{A}^{\ensuremath{'}}}}_{i},{{\ensuremath{\psi}}^{A}}_{i})}_{F}$ on a final surface, which is given by a Feynman path integral. In a semiclassical expansion of this amplitude around a classical solution, the constraints imply that the one- and higher-loop terms $A,{A}_{1},{A}_{2},\dots{}$ are invariant under left-handed supersymmetry transformations at the final surface, and under right-handed transformations at the initial surface. An alternative approach to perturbation theory is provided by the multiple-scattering expansion, which constructs higher-order terms from the one-loop approximation $A \mathrm{exp}(\frac{i{S}_{c}}{\ensuremath{\hbar}})$ to $K$, where ${S}_{c}$ is the classical action. This gives a resummation of the standard semiclassical expansion, which may help in improving the convergence of the theory. The invariance of $A$ under left-handed supersymmetry at the final surface is shown to limit the allowed surface divergences in $A$; there is at most one possible surface counterterm at one loop. Similar restrictions on surface counterterms in the standard expansion are expected at higher-loop order; these conditions may possibly also affect the usual volume counterterms, which must here be accompanied by surface contributions.