  Tensor networks (TNs) have become one of the most essential building blocks
for various fields of theoretical physics such as condensed matter theory,
statistical mechanics, quantum information, and quantum gravity. This review
provides a unified description of a series of developments in the TN from the
statistical mechanics side. In particular, we begin with the variational
principle for the transfer matrix of the 2D Ising model, which naturally leads
us to the matrix product state (MPS) and the corner transfer matrix (CTM). We
then explain how the CTM can be evolved to such MPS-based approaches as density
matrix renormalization group (DMRG) and infinite time-evolved block decimation.
We also elucidate that the finite-size DMRG played an intrinsic role for
incorporating various quantum information concepts in subsequent developments
in the TN. After surveying higher-dimensional generalizations like tensor
product states or projected entangled pair states, we describe tensor
renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson
type real-space renormalization groups, focusing on their fixed point
structures. We then discuss how the difficulty in TRGs for critical systems can
be overcome in the tensor network renormalization and the multi-scale
entanglement renormalization ansatz.
