Delone sets are uniformly discrete point sets X in Euclidean d-space used in the modeling of crystals. They are characterized by two parameters r and R, where (usually) 2r is the smallest inter-point distance of X and R is the radius of a largest "empty ball" that can be placed into the interstices of X. The local theory for Delone sets searches for local conditions on X that guarantee the emergence of a crystallographic group of symmetries producing X as an orbit set consisting of a single point orbit or finitely many point orbits, respectively. The regularity radius t(d) is defined as the smallest positive number s such that each Delone set X with congruent clusters of radius t in Euclidean d-space is a regular system, that is, a point orbit under a crystallographic group. We discuss bounds for the regularity radius in terms of R, and present conjectures that have been verified for some particularly interesting classes of Delone sets. This is joint work with Nikolai Dolbilin, Alexey Garber and Marjorie Senechal.