Implementing Markov's Limiting Curvature Hypothesis

The effective action for gravity at high curvatures is likely to contain higher derivative terms. These corrections may have profound consequences for the singularity structure of space-time and for early Universe cosmology. In this contribution, recent work is reviewed which demonstrates that it is possible to construct a class of effective gravitational actions for which all solutions with sufficient symmetries have limited curvature and are nonsingular. Near the limiting curvature, the coupling between matter and gravity goes to zero and in this sense the theory is asymptotically free.