  Regression problems are traditionally analyzed via univariate characteristics
like the regression function, scale function and marginal density of regression
errors. These characteristics are useful and informative whenever the
association between the predictor and the response is relatively simple. More
detailed information about the association can be provided by the conditional
density of the response given the predictor. For the first time in the
literature, this article develops the theory of minimax estimation of the
conditional density for regression settings with fixed and random designs of
predictors, bounded and unbounded responses and a vast set of anisotropic
classes of conditional densities. The study of fixed design regression is of
special interest and novelty because the known literature is devoted to the
case of random predictors. For the aforementioned models, the paper suggests a
universal adaptive estimator which (i) matches performance of an oracle that
knows both an underlying model and an estimated conditional density; (ii) is
sharp minimax over a vast class of anisotropic conditional densities; (iii) is
at least rate minimax when the response is independent of the predictor and
thus a bivariate conditional density becomes a univariate density; (iv) is
adaptive to an underlying design (fixed or random) of predictors.
