In these lecture notes we give a technical overview of tangent-space methods
for matrix product states in the thermodynamic limit. We introduce the manifold
of uniform matrix product states, show how to compute different types of
observables, and discuss the concept of a tangent space. We explain how to
variationally optimize ground-state approximations, implement real-time
evolution and describe elementary excitations for a given model Hamiltonian.
Also, we explain how matrix product states approximate fixed points of
one-dimensional transfer matrices. We show how all these methods can be
translated to the language of continuous matrix product states for
one-dimensional field theories. We conclude with some extensions of the
tangent-space formalism and with an outlook to new applications.