Summary: In recent years, there has been considerable interest in estimating conditional independence graphs in high dimensions. Most previous work assumed that the variables are multivariate Gaussian or that the conditional means of the variables are linearly related. Unfortunately, if these assumptions are violated, the resulting conditional independence estimates can be inaccurate. We propose a semiparametric method, graph estimation with joint additive models, which allows the conditional means of the features to take an arbitrary additive form. We present an efficient algorithm for computation of our estimator, and prove that it is consistent. We extend our method to estimation of directed graphs with known causal ordering. Using simulated data, we show that our method performs better than existing methods when there are nonlinear relationships among the features, and is comparable to methods that assume multivariate normality when the conditional means are linear. We illustrate our method on a cell signalling data set.