Consider a stationary Poisson process $η$ in a $d$-dimensional hyperbolic space of constant curvature $-\varkappa$ and let the points of $η$ together with a fixed origin $o$ be the vertices of a graph. Connect each point $x\inη$ with its radial nearest neighbour, which is the hyperbolically nearest vertex to $x$ that is closer to $o$ than $x$. This construction gives rise to the hyperbolic radial spanning tree, whose geometric properties are in the focus of this paper. In particular, the degree of the origin is studied. For increasing balls around $o$ as observation windows, expectation and variance asymptotics as well as a quantitative central limit theorem for a class of edge-length functionals are derived. The results are contrasted with those for the Euclidean radial spanning tree.