We survey some recent results about operators on the Bergman space, related to the following two questions: when does the boundary vanishing of the Berezin transform determine compactness of the operator, and how can one determine if a general operator is in the Toeplitz algebra or not? We present an interesting variety of results that provide some particular answers. Moreover, they also reveal connections between these two questions and, even more generally, display deep connections between mathematical areas such as complex analysis, measure theory, and operator theory.