Nonparametric regression with a doubly truncated response is introduced. Local constant and local linear kernel-type estimators are proposed. Asymptotic expressions for the bias and the variance of the estimators are obtained, showing the deterioration provoked by the random truncation. To solve the crucial problem of bandwidth choice, two different bandwidth selectors based on plug-in and cross-validation ideas are introduced. The performance of both the estimators and the bandwidth selectors is investigated through simulations. A real data illustration is included. The main conclusion is that the introduced regression methods perform satisfactorily in the complicated scenario of random double truncation.