The reduced modular Jacobi group is a semidirect product of $\mathrm{SL}_2(\mathbb{Z})$ with the lattice $\mathbb{Z}^2$. We develop the spectral theory of the invariant Laplacian $L$ on the associated group manifold. The operator $L$ is decomposed by Fourier analysis as a direct sum of operators $L_{kl}$ corresponding to frequencies $k$ related to the lattice and $l$ to translations. $L_{00}$ is the Selberg Laplacian for $\mathrm{SL}_2(\mathbb{Z})$. For $k,l\geq 1$, $L_{kl}$ has a purely discrete spectrum, while $L_{k0}$ has a purely continuous spectrum for $k\geq1$. The set of all eigenvalues of $L$ satisfies a Weyl law. The results are extended to subgroups of the modular Jacobi group of finite index.