Koopmans-compliant functionals emerge naturally from extending the constraint of piecewise linearity of the total energy as a function of the number of electrons to each fractional orbital occupation. When applied to approximate density-functional theory, these corrections give rise to orbital-density-dependent functionals and potentials. We show that the simplest implementations of Koopmans' compliance provide accurate estimates for the quasiparticle excitations and leave the total energy functional almost or exactly intact, i.e., they describe correctly electron removals or additions, but do not necessarily alter the electronic charge density distribution within the system. Additional Koopmans-compliant functionals can be constructed that modify the potential energy surface, starting, e.g., from Perdew-Zunger corrections. These functionals become exactly one-electron self-interaction free and, as all Koopmans-compliant functionals, are approximately many-electron self-interaction free. We discuss in detail these different formulations, and provide extensive benchmarks for the 55 molecules in the referenceG2-1 set, using Koopmans-compliant functionals constructed from local-density or generalized-gradient approximations. In all cases, we find excellent performance in the electronic properties, comparable or improved with respect to that of many-body perturbation theories, such as G(0)W(0) and self-consistent GW, at a fraction of the cost and in a variational framework that also delivers energy derivatives. Structural properties and atomization energies preserve or slightly improve the accuracy of the underlying density-functional approximations.