  We study $d$-wave superconductivity in twisted bilayer graphene and reveal
phenomena that arise due to the moir\'e superlattice. In the $d$-wave pairing,
the relative motion (RM) of two electrons in a Cooper pair can have either
$d+id$ or $d-id$ symmetry with opposite angular momenta. Due to the enlarged
moir\'e superlattice, the center-of-mass motion (COMM) can also carry a finite
angular momentum while preserving the moir\'e periodicity. By matching the
total angular momentum, which has contributions from both the RM and the COMM,
Cooper pairs with $d+id$ and $d-id$ RMs are intrinsically coupled in a way such
that the COMM associated with one of the RMs has a spontaneous
vortex-antivortex lattice configuration. Another phenomenon is that the chiral
$d$-wave state carries spontaneous bulk circulating supercurrent. The chiral
$d$-wave superconductors are gapped and also topological as characterized by an
integer Chern number. Nematic $d$-wave superconductors, which could be
stabilized, for example, by uniaxial strain, are gapless with point nodes.
