Multivariate Gaussian data are completely characterized by their mean and covariance but higher-order cumulants are unavoidable in non-Gaussian data. For univariate data, these are well-studied via skewness and kurtosis but for multivariate data, these cumulants are tensor-valued --- higher-order analogs of the covariance matrix capturing higher-order dependence in the data. We argue that multivariate cumulants may be studied via their principal components, defined in a manner analogous to the usual principal components of a covariance matrix. It is best viewed as a subspace selection method that accounts for higher-order dependence the way PCA obtains varimax subspaces. A variant of stochastic gradient descent on the Grassmannian permits us to estimate principal components of cumulants of any order in excess of 10,000 dimensions readily on a laptop computer. This is joint work with Jason Morton.