On Topologies and Boundaries in Potential Theory : Enlarged edition of a course of lectures delivered in 1966 General notions of thinness and fine topology Notion of reduced function. Applications. Strong thinness and strong unthinness General results on fine limits Quasi topological notions Weak thinness Notions in classical potential theory Classical fine topology general properties Applications to balayage weights and capacities Further study of classical thinness. Some applications Relations with the Choquet boundary Extension to axiomatic theories of harmonic functions Abstract minimal thinness minimal boundary minimal fine topology General compactification of constantinescu cornea first examples of application Classical martin space the martin integral representation Classical martin space and minimal thinness Classical martin boundary dirichlet problem and boundary behaviour Comparison of both thinnesses. Fine limits and non tangential limits. (Classical case. Examples) Martin space and minimal thinness in axiomatic theories — short survey.