Over a series of four articles and an introduction, a "method of pseudo-moments" is developed which gives polynomial-time tensor rank decompositions for a variety of tensor component models. Algorithms given fall into two general classes: those falling back on convex optimization, which develop the theory of polynomial-time algorithms, as well as those constructed through spectral and matrix polynomial methods, which illustrate the possibility of realizing the aforementioned polynomial-time algorithm ideas in runtimes practical for real-life inputs. Tensor component models covered include those featuring random, worst-case, generic, or smoothed inputs, as well as cases featuring overcomplete and undercomplete rank. All models are capable of tolerating substantial noise in tensor inputs, measured in spectral norm.