This thesis is concerned with the problem of performing causal graphical structure learning. The unifying approach to the problems studied throughout the thesis is the use of nonparametric machine learning techniques in order to relax distributional and functional assumptions on the data generating processes under consideration. The contribution of the thesis are four distinct manuscripts that are each concerned with different aspects of structure learning, which can be divided into two overall themes. The first theme is structure learning of graphical models for multivariate time series. Here we consider detecting the edges of a graphical model by posing regression models of the time series and reading the graph structure off the fitted models. The second theme is nonparametric hypothesis tests for constraint-based structure learning. Here we develop novel tests for conditional independence and conditional local independence. Our test for conditional independence is based on a generalized correlation in the partial copula, where we estimate nonparametric residuals using quantile regression. Our test for conditional local independence is based on a stochastic integral, which is a zero-mean local martingale under the hypothesis, and where the test statistic process requires nonparametric estimation of an intensity function and a predictable projection process. For both tests we utilize techniques from double machine learning to perform inference on a test statistic of a dependence measure in the presence of infinite dimensional nuisance parameters.