Quantile regression has emerged as a significant extension of traditional linear models and its potential in survival applications has recently been recognized. In this paper we study quantile regression with competing risks data, formulating the model based on conditional quantiles defined using the cumulative incidence function, which includes as a special case an analog to the usual accelerated failure time model. The proposed competing risks quantile regression model provides meaningful physical interpretations of covariate effects and, moreover, relaxes the constancy constraint on regression coefficients, thereby providing a useful, perhaps more flexible, alternative to the popular subdistribution proportional hazards model. We derive an unbiased monotone estimating equation for regression parameters in the quantile model. The uniform consistency and weak convergence of the resulting estimators are established across a quantile continuum. We develop inferences, including covariance estimation, second-stage exploration, and model diagnostics, which can be stably implemented using standard statistical software without involving smoothing or resampling. Our proposals are illustrated via simulation studies and an application to a breast cancer clinical trial.