Several research thrusts in the area of data management have focused on understanding how changes in the data affect the output of a view or standing query. Example applications are explaining query results, propagating updates through views, and anonymizing datasets. These applications usually rely on understanding how interventions in a database impact the output of a query. An important aspect of this analysis is the problem of deleting a minimum number of tuples from the input tables to make a given Boolean query false. We refer to this problem as "the resilience of a query" and show its connections to the well-studied problems of deletion propagation and causal responsibility. In this paper, we study the complexity of resilience for self-join-free conjunctive queries, and also make several contributions to previous known results for the problems of deletion propagation with source side-effects and causal responsibility: (1) We define the notion of resilience and provide a complete dichotomy for the class of self-join-free conjunctive queries with arbitrary functional dependencies; this dichotomy also extends and generalizes previous tractability results on deletion propagation with source side-effects. (2) We formalize the connection between resilience and causal responsibility, and show that resilience has a larger class of tractable queries than responsibility. (3) We identify a mistake in a previous dichotomy for the problem of causal responsibility and offer a revised characterization based on new, simpler, and more intuitive notions. (4) Finally, we extend the dichotomy for causal responsibility in two ways: (a) we treat cases where the input tables contain functional dependencies, and (b) we compute responsibility for a set of tuples specified via wildcards.