In this paper, we rigorously study tractable models for provably recovering low-rank tensors. Unlike their matrix-based predecessors, current convex approaches for recovering low-rank tensors based on incomplete (tensor completion) and/or grossly corrupted (tensor robust principal analysis) observations still suffer from the lack of theoretical guarantees, although they have been used in various recent applications and have exhibited promising empirical performance. In this work, we attempt to fill this gap. Specifically, we propose a class of convex recovery models (including strongly convex programs) that can be proved to guarantee exact recovery under certain conditions. All parameters in our formulations can be determined beforehand based on the measurement data and thus there is no parameter tuning involved.