Space partitioning methods such as random forests and the Mondrian process are powerful machine learning methods for multi-dimensional and relational data, and are based on recursively cutting a domain. The flexibility of these methods is often limited by the requirement that the cuts be axis aligned. The Ostomachion process and the self-consistent binary space partitioning-tree process were recently introduced as generalizations of the Mondrian process for space partitioning with non-axis aligned cuts in the plane. Motivated by the need for a multi-dimensional partitioning tree with non-axis aligned cuts, we propose the Random Tessellation Process, a framework that includes the Mondrian process as a special case. We derive a sequential Monte Carlo algorithm for inference, and provide random forest methods. Our methods are self-consistent and can relax axis-aligned constraints, allowing complex inter-dimensional dependence to be captured. We present a simulation study and analyze gene expression data of brain tissue, showing improved accuracies over other methods.