A substantial progress in development of new and efficient tensor factorization techniques has led to an extensive research of their applicability in recommender systems field. Tensor-based recommender models push the boundaries of traditional collaborative filtering techniques by taking into account a multifaceted nature of real environments, which allows to produce more accurate, situational (e.g. context-aware, criteria-driven) recommendations. Despite the promising results, tensor-based methods are poorly covered in existing recommender systems surveys. This survey aims to complement previous works and provide a comprehensive overview on the subject. To the best of our knowledge, this is the first attempt to consolidate studies from various application domains, which helps to get a notion of the current state of the field. We also provide a high level discussion of the future perspectives and directions for further improvement of tensor-based recommendation systems.