  Coupled cluster theory is the method of choice for weakly correlated systems.
But in the strongly correlated regime, it faces a symmetry dilemma, where it
either completely fails to describe the system, or has to artificially break
certain symmetries. On the other hand, projected Hartree-Fock theory captures
the essential physics of many kinds of strong correlations via symmetry
breaking and restoration. In this work, we combine and try to retain the merits
of these two methods by applying symmetry projection to broken symmetry coupled
cluster wavefunctions. The non-orthogonal nature of states resulting from the
application of symmetry projection operators furnishes particle-hole
excitations to all orders, thus creating an obstacle for the exact evaluation
of overlaps. Here we provide a solution via a disentanglement framework theory
that can be approximated rigorously and systematically. Results of projected
coupled cluster theory are presented for molecules and the Hubbard model,
showing that spin projection significantly improves unrestricted coupled
cluster theory while restoring good quantum numbers. The energy of projected
coupled cluster theory reduces to the unprojected one in the thermodynamic
limit, albeit at a much slower rate than projected Hartree-Fock.
