  Motivated by the recent proposals for unconventional emergent physics in
twisted bilayers of nodal superconductors, we study the peculiarities of the
Josephson effect at the twisted interface between $d$-wave superconductors. We
demonstrate that for clean interfaces with a twist angle $\theta_0$ in the
range $0^\circ<\theta_0<45^\circ$ the critical current can exhibit nonmonotonic
temperature dependence with a maximum at a nonzero temperature as well as a
complex dependence on the twist angle at low temperatures. The former is shown
to arise quite generically due to the contributions of the momenta around the
gap nodes, which are negative for nonzero twist angles. It is demonstrated that
these features reflect the geometry of the Fermi surface and are sensitive to
the form of the momentum dependence of the tunneling at the twisted interface.
Close to $\theta_0=45^\circ$ we find that the critical current does not vanish
due to Cooper pair cotunneling, which leads to a transition to a time-reversal
breaking topological superconducting $d+id$ phase. Weak interface roughness,
quasiperiodicity, and inhomogeneity broaden the momentum dependence of the
interlayer tunneling leading to a critical current $I_c\sim \cos(2\theta_0)$
with $\cos(6\theta_0)$ corrections. Furthermore, strong disorder at the
interface is demonstrated to suppress the time-reversal breaking
superconducting phase near $\theta_0=45^\circ$. Last, we provide a
comprehensive theoretical analysis of experiments that can reveal the full
current-phase relation for twisted superconductors close to
$\theta_0=45^\circ$. In particular, we demonstrate the emergence of the
Fraunhofer interference pattern near $\theta_0=45^\circ$, while accounting for
realistic sample geometries, and show that its temperature dependence can yield
unambiguous evidence of Cooper pair cotunneling, necessary for topological
superconductivity.
