  Deep generative models have shown tremendous capability in data density
estimation and data generation from finite samples. While these models have
shown impressive performance by learning correlations among features in the
data, some fundamental shortcomings are their lack of explainability, tendency
to induce spurious correlations, and poor out-of-distribution extrapolation. To
remedy such challenges, recent work has proposed a shift toward causal
generative models. Causal models offer several beneficial properties to deep
generative models, such as distribution shift robustness, fairness, and
interpretability. Structural causal models (SCMs) describe data-generating
processes and model complex causal relationships and mechanisms among variables
in a system. Thus, SCMs can naturally be combined with deep generative models.
We provide a technical survey on causal generative modeling categorized into
causal representation learning and controllable counterfactual generation
methods. We focus on fundamental theory, methodology, drawbacks, datasets, and
metrics. Then, we cover applications of causal generative models in fairness,
privacy, out-of-distribution generalization, precision medicine, and biological
sciences. Lastly, we discuss open problems and fruitful research directions for
future work in the field.
