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Transfer matrices and matrix product operators play an ubiquitous role in thefield of many body physics. This paper gives an ideosyncratic overview ofapplications, exact results and computational aspects of diagonalizing transfermatrices and matrix product operators. The results in this paper are a mixtureof classic results, presented from the point of view of tensor networks, and ofnew results. Topics discussed are exact solutions of transfer matrices inequilibrium and non-equilibrium statistical physics, tensor network states,matrix product operator algebras, and numerical matrix product state methodsfor finding extremal eigenvectors of matrix product operators.

Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods