Despite the success and the formal elegance, general relativity suffers from shortcomings at any scale of energy. For this reason, several alternatives have been proposed over the years, with the aim to address issues provided by Einstein's gravity. Within the plethora of modified theories of gravity, particular interest is gained by modifications including topological invariants, such as the Gauss-Bonnet term. Extending the gravitational action with the Gauss-Bonnet invariant leads to interesting dynamics in cosmological and spherically symmetric backgrounds. Specifically, the early and late-time Universe bahaviours can be predicted without invoking any Dark Energy, which is thus mimicked by geometric contributions. Moreover, considering the Gauss-Bonnet term into the action can settle theoretical issues, currently faced by General Relativity. After selecting the form of the action, the gravitational field equations can yield analytic solutions whose free parameters can be constrained by experiments and observations. Specifically, the starting model can be selected by the so called "Noether Symmetry Approach", a selection criterion aimed at finding easily-handled models containing symmetries.

The resulting conserved quantities can be used to reduce the minisuperspace dimension and find out exact solutions. In this talk, different modifications of General Relativity including the Gauss-Bonnet term are discussed, such as scalar-tensor models, non-local models, extended and teleparallel models.