A wide variety of efficient algorithms have been developed for MPS/TT tensor networks.
The following algorithms involve solving equations such as $A x = \lambda x$ or $A x = b$ where $x$ is a tensor in MPS/TT form.
The following are algorithms for summing two or more MPS/TT networks and approximating the result by a single MPS/TT.
The following are algorithms for multiplying a given MPS/TT tensor network by an MPO tensor network, resulting in a new MPS/TT that approximates the result.
One reason MPS are very useful in quantum physics applications is that they can be efficiently evolved in real or imaginary time. This capability is useful for studying quantum dynamics and thermalization, and directly simulating finite-temperature systems.