  We show how to construct relevant families of matrix product operators in one
and higher dimensions. Those form the building blocks for the numerical
simulation methods based on matrix product states and projected entangled pair
states. In particular, we construct translational invariant matrix product
operators suitable for time evolution, and show how such descriptions are
possible for Hamiltonians with long-range interactions. We illustrate how those
tools can be exploited for constructing new algorithms for simulating quantum
spin systems.
