For a given square matrix $A$, $q(A)$ denotes the number of distinct eigenvalues of $A$. For a given graph $G$, and $S(G)$ the set of all real symmetric matrices associated with $G$, we define $q(G)$ as the minimum of $q(A)$ over all $A$ in $S(G)$. We refer to $q(G)$ as the minimum number of distinct eigenvalues associated with $G$. In this talk, a survey of existing facts and properties of $q(G)$ will be presented along with some new results on $q(G)$. In particular, we will discuss extreme values of $q(G)$ and connections to some related parameters that are monotone on subgraphs.