Hankel and Toeplitz operators defined on Bergman spaces are an active area of research in complex analysis and operator theory. The compactness of these operators has been extensively explored in many studies. In earlier works, the compactness of these operators was linked to various conditions on the symbol that generates the operator. Čučković and Şahutoğlu shown that the compactness of Hankel and Toeplitz operators defined on Bergman spaces is closely related to the complex geometry of the region where the Bergman space is defined, using techniques from several complex variables. In this talk, I will introduce the results obtained using both approaches and the techniques employed. I will present the results obtained for a more general case in which neither technique was used to characterize the compactness of Hankel and Toeplitz operators. This is joint work with Sönmez Şahutoğlu from the University of Toledo.