  We discuss how the internal structure of ultracold molecules, trapped in the
motional ground state of optical tweezers, can be used to implement qudits. We
explore the rotational, fine and hyperfine structure of $^{40}$Ca$^{19}$F and
$^{87}$Rb$^{133}$Cs, which are examples of molecules with $^2\Sigma$ and
$^1\Sigma$ electronic ground states, respectively. In each case we identify a
subset of levels within a single rotational manifold suitable to implement a
4-level qudit. Quantum gates can be implemented using two-photon microwave
transitions via levels in a neighboring rotational manifold. We discuss
limitations to the usefulness of molecular qudits, arising from off-resonant
excitation and decoherence. As an example, we present a protocol for using a
molecular qudit of dimension $d=4$ to perform the Deutsch algorithm.
