We investigate two novel mixed robust/average-case submodular data partitioning problems that we collectively call \emph{Submodular Partitioning}. These problems generalize purely robust instances of the problem, namely \emph{max-min submodular fair allocation} (SFA) and \emph{min-max submodular load balancing} (SLB), and also average-case instances, that is the \emph{submodular welfare problem} (SWP) and \emph{submodular multiway partition} (SMP). While the robust versions have been studied in the theory community, existing work has focused on tight approximation guarantees, and the resultant algorithms are not generally scalable to large real-world applications. This is in contrast to the average case, where most of the algorithms are scalable. In the present paper, we bridge this gap, by proposing several new algorithms (including greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large datasets but that also achieve theoretical approximation guarantees comparable to the state-of-the-art. We moreover provide new scalable algorithms that apply to additive combinations of the robust and average-case objectives. We show that these problems have many applications in machine learning (ML), including data partitioning and load balancing for distributed ML, data clustering, and image segmentation. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization (of convex and deep neural network objectives), and also purely unsupervised image segmentation.