A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$ and for $d \geq 4$ such that $d \equiv 0 \pmod{4}$. Here it is shown that infinitely many $d$-regular nut graphs exist for each degree $d \geq 3$. Moreover, we prove that there are infinitely many $d$-regular Cayley nut graphs for each even $d \ge 4$. This implies that we have identified all feasible degrees $d$ for which a $d$-regular Cayley nut graph exists.