This is part of our ongoing efforts to understand high-dimensional combinatorial objects. These include simplicial complexes (a 1-dimensional simpicial complex is a graph), high-dimensional permutations (e.g., a 2-dimensional permutation is synonymous with a Latin square), hypertrees and more. For every type of combinatorial object that we wish to study it is crucial to understand their typical behavior. Namely, to understand the properties of random objects of the type that we study. In this talk I describe recent work with my student Maya Dotan which allows us to efficiently generate random one-factorizations. I will also mention ongoing work with my student Michael Simkin on random generation of Latin squares. In addition I will speak about joint work with Zur Luria on low-discrepancy high-dimensional permutations.