Chemical graph theory (CGT) is a branch of mathematical chemistry which deals with the nontrivial applications of graph theory to solve molecular problems. In general, a graph is used to represent a molecule by considering the atoms as the vertices of the graph and the molecular bonds as the edges. Then, the main goal of CGT is to use algebraic invariants to reduce the topological structure of a molecule to a single number which characterizes either the energy of the molecule as a whole or its orbitals, its molecular branching, structural fragments, and its electronic structures, among others. These graph theoretic invariants are expected to correlate with physical observable measures by experiments in a way that theoretical predictions can be used to gain chemical insights even for not yet existing molecules. In this brief review we shall present a selection of results in some of the most relevant areas of CGT.