  We study superconductors with $n$-fold rotational invariance both in the
presence and in the absence of spin-orbit interactions. More specifically, we
classify the non-interacting Hamiltonians by defining a series of $Z$-numbers
for the Bogoliubov-de Gennes (BdG) symmetry classes of the Altland-Zimbauer
classification of random matrices in $1$D, $2$D, and $3$D in the presence of
discrete rotational invariance. Our analysis emphasizes the important role
played by the angular momentum of the Cooper pairs in the system: for pairings
of nonzero angular momentum, the rotation symmetry may be represented
projectively, and a projective representation of rotation symmetry may have
anomalous properties, including the anti-commutation with the time-reversal
symmetry. In 1D and 3D, we show how an $n$-fold axis enhances the topological
classification and give additional topological numbers; in 2D, we establish a
relation between the Chern number (in class D and CI) and the eigenvalues of
rotation symmetry at high-symmetry points. For each nontrivial class in 3D, we
write down a minimal effective theory for the surface Majorana states.
