Algebraic Geometry is an old subject in mathematics and, at the same time, a vibrant area of current research with close connections to other areas. Its main objects of study are algebraic varieties which means, roughly speaking, zero sets of of polynomials. For example, an algebraic curve is a 1-dimensional algebraic variety, and an algebraic surface is a 2-dimensional algebraic variety. Algebraic invariants of varieties provide us with a way of understanding how these geometric objects look, in a sense. These invariants can be of both topological and geometric nature and, generally, associate numbers to geometric phenomena. From this point of view the the main objectives of the project are two-fold: on the one hand, we try to understand if algebraic varieties with given numerical invariants exist and, on the other hand, given an algebraic variety, we wish to understand its numerical invariants. With respect to the first objective, our focus is mainly on algebraic surfaces. A classification of these does exist but, nevertheless, there are many open problems concerning both existence of certain types of surface and also their geometric properties, which we shall throw light on. With respect to the second objective we focus on moduli spaces of Higgs bundles and quiver bundles. These are central objects in current geometry: indeed Higgs bundle moduli play an important role in hot topics like the geometric Langlands programme and mirror symmetry. Moreover, they are related to character varieties for surface groups through the Non-abelian Hodge Theorem. We shall improve the understanding of the geometry and topology of these moduli spaces and explore their relevance in the aforementioned contexts.