In this talk, we will discuss several tensor optimization problems that require the resulting solution to have a low-rank structure. These problems include low-rank tensor completion, low-rank and sparse tensor separation and tensor principal component analysis. We show that although these problems are all nonconvex and NP-hard in general, they can be approximated very well in practice by certain structured convex optimization problems. Numerical results on applications arising from computer vision and portfolio selection with higher-order moments will be presented.