The study of Hankel and Toeplitz operators on spaces of holomorphic functions has been a central theme in operator theory. In this talk, we focus on these operators when they are defined on Bergman spaces over bounded domains in Cn. After providing the necessary background and definitions, we will concentrate on the compactness of these operators. A key part of the discussion will be devoted to the functional analytic and several complex variable techniques that are employed to analyze compactness. In particular, we will investigate the compactness of Hermitian squares of Hankel and Toeplitz operators, which arise naturally in various contexts. The talk will conclude with recent joint results obtained with Sönmez Şahutoğlu from the University of Toledo.