Tensors are multiway arrays, and tensor decompositions are powerful tools for data analysis and compression. In this talk, we demonstrate the wide-ranging utility of both the canonical polyadic (CP) and Tucker tensor decompositions with examples in neuroscience, social networks, and combustion science. We explain the model-fitting challenges for CP, including nonconvexity and NP-hardness, as well as the benefits, including uniqueness of the decomposition and the interpretability the results. We discuss the different types of tensor decompositions. For instance, a different choice of the fit metric in CP leads to Poisson Tensor Factorization for count data. Tucker has several advantages compared to CP such as the ability to easily compute the rank and even the rank required for a specific level of approximation. We present new results in scalability for both methods. For CP, we present a novel randomization method that not only improves the speed of the computation but also its robustness. For Tucker, we present results on compressing massive data sets by orders of magnitude by discovery of latent low-dimensional manifolds.