Riemannian geometry deals, roughly speaking, with manifolds covered by a model space whose geometric structure is important per se. In (real) dimension $4$, the complex hyperbolic plane is a basic and fundamental model space. We will describe a series of complex hyperbolic disc bundles over closed surfaces that arise from certain discrete groups generated by reflections in complex geodesics in the complex hyperbolic plane. The emphasis will be on the interplay between topology and geometry; for example, we will explain how the Euler number of a bundle can be obtained from certain fractional Euler numbers. Some of our results answer known problems in complex hyperbolic geometry. No previous knowledge of complex hyperbolic geometry is assumed. We will also give a brief and elementary introduction to the topology of disc bundles over closed surfaces. This is part of a joint work with Sasha Ananin (UNICAMP, Brazil) and Nikolay Gusevskii (UFMG, Brazil).