There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot–Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms.

The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting.

We are given a metric space \(X\) equipped with a doubling measure \(\mu\). A generalization of a Sobolev function and its gradient is a pair \(u\in L^{1}_{\rm loc}(X)\), \(0\leq g\in L^{p}(X)\) such that for every ball \(B\subset X\) the Poincaré-type inequality \[ ⨍_{B} |u-u_{B}|\, d\mu \leq C r ( ⨍_{\sigma B} g^{p}\, d\mu)^{1/p}\,\] holds, where \(r\) is the radius of \(B\) and \(\sigma\geq 1\), \(C>0\) are fixed constants. Working in the above setting we show that basically all relevant results from the classical theory have their counterparts in our general setting. These include Sobolev-Poincaré type embeddings, Rellich-Kondrachov compact embedding theorem, and even a version of the Sobolev embedding theorem on spheres. The second part of the paper is devoted to examples and applications in the above mentioned areas.