Traditionally in the social sciences, causal mediation analysis has been formulated, understood, and implemented within the framework of linear structural equation models. We argue and demonstrate that this is problematic for three reasons; the lack of a general definition of causal mediation effects independent of a particular statistical model, the inability to specify the key identification assumption, and the difficulty of extending the framework to nonlinear models. In this paper, we propose an alternative approach that overcomes these limitations. Our approach is general because it offers the definition, identification, estimation, and sensitivity analysis of causal mediation effects without reference to any specific statistical model. Further, our approach explicitly links these four elements closely together within a single framework. As a result, the proposed framework can accommodate linear and nonlinear relationships, parametric and nonparametric models, continuous and discrete mediators, and various types of outcome variables. The general definition and identification result also allow us to develop sensitivity analysis in the context of commonly used models, which enables applied researchers to formally assess the robustness of their empirical conclusions to violations of the key assumption. We illustrate our approach by applying it to the Job Search Intervention Study (JOBS II). We also offer easy-to-use software that implements all of our proposed methods.