Cayley graphs provide interesting bridges between graph theory, additive combinatorics and group theory. Fixing an ambient finite group, random Cayley graphs are constructed by choosing a generating set at random. These graphs reflect interesting symmetries and properties of the group, at the cost of inducing complex dependencies. Recent progress in the analysis of cliques and independent sets in random Cayley graphs has brought about interesting lessons and surprises. For instance, our analysis of Ramsey properties of dense random Cayley graphs has motivated a purely combinatorial approach to questions in additive combinatorics, which subsequently led to the resolution of a conjecture of Ruzsa in additive combinatorics. In this talk, I will discuss further developments in the sparse regime. The study of independence number of sparse random Cayley graphs has interesting connections to a question of Ben Green on the largest subset of an abelian group that cannot be written as a sumset. The analysis of sparse random Cayley graphs is significantly more intricate due to failure of the first moment method. Instead, a key ingredient is the construction of efficient `covers’ for sumsets, which we obtain from combining information from both the physical and Fourier space. Along the way, we will also need several robust results in additive combinatorics, which we obtain from the combinatorial perspective discovered in the dense case.