The cosmological constant problem and the compatibility of gravity with quantum mechanics are the two most pressing problems in all of gravitational theory. While string theory nicely addresses the latter, it has so far failed to provide any compelling solution to the former. On the other hand, while conformal gravity nicely addresses the cosmological constant problem [by naturally quenching the amount by which the cosmological constant gravitates rather than by quenching the cosmological constant itself -- Mannheim, Prog. Part. Nuc. Phys. 56, 340 (2006)], the fourth order derivative conformal theory has long been thought to possess a ghost when quantized. However, it has recently been shown by Bender and Mannheim [Phys. Rev. Lett. 100, 110402 (2008)] that not only do theories based on fourth order derivative equations of motion not have ghosts, they actually never had any to begin with, with the apparent presence of ghosts being due entirely to treating operators which were not Hermitian on the real axis as though they were. When this is taken care of via an underlying PT symmetry that such theories are found to possess, there are then no ghosts at all and the theory is fully unitary. Conformal gravity is thus advanced as a fully consistent four-dimensional alternative to ten-dimensional string theory.