The general themes of the talk are Dirichlet forms, fractional operators and associated Sobolev type spaces on groups of homogeneous type. Our results lead to explicit integral formulas of the infinitesimal generators, which are the studied fractional operators, and embedding theorems between the relevant spaces. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type conditions. Finally, we will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.