  We investigate the homology of ample Hausdorff groupoids. We establish that a
number of notions of equivalence of groupoids appearing in the literature
coincide for ample Hausdorff groupoids, and deduce that they all preserve
groupoid homology. We compute the homology of a Deaconu{Renault groupoid
associated to k pairwisecommuting local homeomorphisms of a zero-dimensional
space, and show that Matui's HK conjecture holds for such a groupoid when k is
one or two. We specialise to k-graph groupoids, and show that their homology
can be computed in terms of the adjacency matrices, using a chain complex
developed by Evans. We show that Matui's HK conjecture holds for the groupoids
of single vertex k-graphs which satisfy a mild joint-coprimality condition. We
also prove that there is a natural homomorphism from the categorical homology
of a k-graph to the homology of its groupoid.
