A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no zero elements. The problem of determining the orders $n$ for which $d$-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for $d \leq 4$. Here we solve the problem for all remaining cases $d\leq 11$ and determine the complete lists of all $d$-regular nut graphs of order $n$ for small values of $d$ and $n$. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any $d$-regular nut graph of order $n$, another $d$-regular nut graph of order $n + 2d$. If we are given a sufficient number of $d$-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all $d$-regular nut graphs of higher orders is established. For even $d$ the orders $n$ are indeed consecutive, while for odd $d$ the orders $n$ are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.