Conditional distribution functions are important statistical objects for the analysis of a wide class of problems in econometrics and statistics. We propose ﬂexible Gaussian representations for conditional distribution functions and give a concave likelihood formulation for their global estimation. We obtain solutions that satisfy the monotonicity property of conditional distribution functions, including under general misspeciﬁcation and in ﬁnite samples. A Lasso-type penalized version of the corresponding maximum likelihood estimator is given that expands the scope of our estimation analysis to models with sparsity. Inference and estimation results for conditional distribution, quantile and density functions implied by our representations are provided and illustrated with an empirical example and numerical simulations.