The sign problem has prevented the use of lattice field theory to many physics systems, including the QCD-based study of nuclear/quark matter. We introduce an approach to bypass the problem - closely related to the "Lefschetz thimble" approach - involving the complexification of the field variables and a deformation of the region of integration. We  discuss conceptual and  algorithmic issues appearing in this approach and apply it to simple models in quantum mechanics and field theories in small lattices.