  Skew-gentle algebras are a generalisation of the well-known class of gentle
algebras with which they share many common properties. In this work, using
non-commutative Gr\"obner basis theory, we show that these algebras are Koszul
and that the Koszul dual is again skew-gentle. We give a geometric model of
their bounded derived categories in terms of polygonal dissections of orbifold
surfaces establishing a correspondence between curves in the orbifold and
indecomposable objects. Moreover, we show that the orbifold dissections encode
homological properties of skew-gentle algebras such as their singularity
categories, their Gorenstein dimensions and derived invariants such as the
determinant of their q-Cartan matrices.
