This thesis develops and assesses new ways to study the conditional quantiles of a population using a sample of data that represents the population. All methods presented here build on a recently-proposed non-parametric approach to quantile regression that is analogous to local linear regression in the least-squares setting. A major challenge is that the raw local quantile estimates are cumbersome to interpret and gain insight from directly. Aiming to overcome this challenge, there are four main contributions herein. First, we demonstrate how a low-rank additive regression analysis can produce insight into a collection of local nonparametric quantile estimates. The low rank structure regularizes the noisy quantile estimates and facilitates interpretation of the findings. Second, we show how a multivariate dimension reduction approach provides a different type of insight into a collection of estimated conditional quantile functions. The third contribution of the thesis leverages the combination of nonparametric quantile estimation and low-rank regression in the context of mediation analysis. We show that this produces a novel quantile-based approach to mediation analysis that expresses direct and indirect effects in a concise and interpretable way. The final methodological contribution of the thesis is a framework for moment-based estimation of conditional covariance functions for stochastic processes. Throughout the thesis, we motivate our work through analyses looking at the proximal and distal factors predicting human blood pressure.