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This paper studies a possible de nition of Sobolev spaces in abstract metric spaces, and answers in the a rmative the question whether this de nition yields a Banach space. The paper also explores the relationship between this de nition and the Haj lasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Di erent v ersions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of pathdoi:10.4171/rmi/275 fatcat:hfrepcsgxvbgvofzbg7e6bxaj4