We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, de ned on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of de nition. This rules out a recently proposed scenario for the RG description of rst-order phase transitions. On the pathological side, we make rigorous some arguments of Grifths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-de ned, and that the conventional RG description of rst-order phase transitions is not universally valid. For decimation or Kadano transformations applied to the Ising model in dimension d 3, these pathologies occur in a full neighborhood f > 0 ; jhj < ( )g of the low-temperature part of the rstorder phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d 2, the pathologies occur at low temperatures for arbitrary magnetic-eld strength. Pathologies may also occur in the critical region for Ising models in dimension d 4. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures and the possible occurrence of the latter in other situations, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.