  In this paper, local Tb theorems are studied both in the doubling and
non-doubling situation. We prove a local Tb theorem for the class of upper
doubling measures. With such general measures, scale invariant testing
conditions are required (L^{\infty} or BMO). In the case of doubling measures,
we also modify the general non-homogeneous method of proof to yield a new proof
of the local Tb theorem with L^2 type testing conditions.
