  The problems considered in this paper come as a natural continuation of our
program to develop a free analogue of Sz.-Nagy-Foias theory, for row
contractions. The paper is structured as follows:
  Introduction
  Part I. Unitary invariants for n-tuples of operators
  1. Joint numerical radius
  2. Euclidean operator radius
  3. Joint numerical range and spectrum
  4. $\rho$-operator radius
  Part II. Joint operator radii, inequalities, and applications
  5. von Neumann inequalities
  6. Constrained von Neumann inequalities
  7. Multivariable Haagerup-de la Harpe inequalities
  8. Multivariable Fejer inequalities
  References
