  In this note we discuss the geometry of Riemannian surfaces having a discrete
set of singular points. We assume the conformal structure extends through the
singularities and the curvature is integrable. Such points are called
\emph{simple singularities}. We first describe them locally and then globally
using the notion of (real) divisor. We formulate a Gauss-Bonnet formula and
relate it to some asymptotic isoperimetric ratio. We prove a classifications
theorem for flat metrics with simple singularities on a compact surface and
discuss the Berger--Nirenberg Problem on surfaces with a divisor. We finally
discuss the relation with spherical polyhedra.
