Comment: 55 pages. arXiv admin note: text overlap with arXiv:2112.07121This paper develops estimation and inference methods for conditional quantile factor models. We first introduce a simple sieve estimation, and establish asymptotic properties of the estimators under large $N$. We then provide a bootstrap procedure for estimating the distributions of the estimators. We also provide two consistent estimators for the number of factors. The methods allow us not only to estimate conditional factor structures of distributions of asset returns utilizing characteristics, but also to conduct robust inference in conditional factor models, which enables us to analyze the cross section of asset returns with heavy tails. We apply the methods to analyze the cross section of individual US stock returns.