  There are several graphs defined on groups. Among them we consider graphs
whose vertex set consists conjugacy classes of a group $G$ and adjacency is
defined by properties of the elements of conjugacy classes. In particular, we
consider commuting/nilpotent/solvable conjugacy class graph of $G$ where two
distinct conjugacy classes $a^G$ and $b^G$ are adjacent if there exist some
elements $x\in a^G$ and $y\in b^G$ such that $\langle x, y \rangle$ is
abelian/nilpotent/solvable. After a section of introductory results and
examples, we discuss all the available results on connectedness, graph
realization, genus, various spectra and energies of certain induced subgraphs
of these graphs. Proofs of the results are not included. However, many open
problems for further investigation are stated.
