Let and be graphs. The subgraph counting function is defined as the maximum possible number of subgraphs in an -vertex -free graph. This function is a direct generalization of the Turán function as . The systematic study of was initiated by Alon and Shikhelman in 2016 who generalized several classical results in extremal graph theory to the subgraph counting setting. Prior to their paper, a number of individual cases were investigated; a well-known example is the question to determine the maximum number of pentagons in a triangle-free graph. In this talk we will survey results on the function including a number of recent papers. We will also discuss this function’s connection to hypergraph Turán problems.