The paper introduces a multiway tensor generalization of the bigraphical lasso which uses a two-way sparse Kronecker sum multivariate normal model for the precision matrix to model parsimoniously conditional dependence relationships of matrix variate data based on the Cartesian product of graphs. We call this tensor graphical lasso generalization TeraLasso. We demonstrate by using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway co-ordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the bigraphical lasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set respectively. We propose a scalable composite gradient descent algorithm and analyse the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate TeraLasso by using both simulation and experimental data from a meteorological data set, showing that we can accurately estimate precision matrices and recover meaningful conditional dependence graphs from high dimensional complex data sets.