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Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniformhypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may growwith $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in whichsuccessive edges overlap in precisely $\ell$ vertices, and 1-overlappingHamilton cycles are called loose Hamilton cycles.  When $r,s\geq 3$ are fixed integers, we establish a threshold result for theproperty of containing a loose Hamilton cycle. This partially verifies aconjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, wealso find the asymptotic distribution of the number of loose Hamilton cycles in$\mathcal{G}(n,r,s)$.  Finally we prove that for $\ell = 2,\ldots, s-1$ and for $r$ growingmoderately as $n\to\infty$, the probability that $\mathcal{G}(n,r,s)$ has a$\ell$-overlapping Hamilton cycle tends to zero.

A threshold result for loose Hamiltonicity in random regular uniform hypergraphs