In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $α, ν> 0$, which we think of as constant. Roughly speaking $α$ controls the power law exponent of the degree sequence and $ν$ the average degree. Here we show that for every $α< 1/2$ and $ν=ν(α)$ sufficiently small, the model does not contain a perfect matching with high probability, whereas for every $α< 1/2$ and $ν=ν(α)$ sufficiently large, the model contains a Hamilton cycle with high probability.