  We construct a ``pseudo-supersymmetric" fermionic extension of the effective
action of the bosonic string in arbitrary spacetime dimension D. The theory is
invariant under pseudo-supersymmetry transformations up to the quadratic
fermion order, which is sufficient in order to be able to derive Killing spinor
equations in bosonic backgrounds, and hence to define BPS type solutions
determined by a system of first-order equations. The pseudo-supersymmetric
theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet.
This also allows us to construct ``\alpha' corrections'' involving quadratic
curvature terms. An exponential dilaton potential term, associated with the
conformal anomaly for a bosonic string outside its critical dimension, can also
be pseudo-supersymmetrised.
