We propose Deep Residual Mixture Models (DRMMs), a novel deep generative model architecture. Compared to other deep models, DRMMs allow more flexible conditional sampling: The model can be trained once with all variables, and then used for sampling with arbitrary combinations of conditioning variables, Gaussian priors, and (in)equality constraints. This provides new opportunities for interactive and exploratory machine learning, where one should minimize the user waiting for retraining a model. We demonstrate DRMMs in constrained multi-limb inverse kinematics and controllable generation of animations.