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We study compactness and boundedness of embeddings from Sobolev type spaceson metric spaces into $L^q$ spaces with respect to another measure. Theconsidered Sobolev spaces can be of fractional order and some statements allowalso nondoubling measures. Our results are formulated in a general form, usingsequences of covering families and local Poincar\'e type inequalities. We showhow to construct such suitable coverings and Poincar\'e inequalities. Forlocally doubling measures, we prove a self-improvement property fortwo-weighted Poincar\'e inequalities, which applies also to lower-dimensionalmeasures.  We simultaneously treat various Sobolev spaces, such as the Newtonian,fractional Haj\l asz and Poincar\'e type spaces, for rather general measuresand sets, including fractals and domains with fractal boundaries. Byconsidering lower-dimensional measures on the boundaries of such domains, weobtain trace embeddings for the above spaces. In the case of Newtonian spaceswe exactly characterize when embeddings into $L^q$ spaces with respect toanother measure are compact. Our tools are illustrated by concrete examples.For measures satisfying suitable dimension conditions, we recover severalclassical embedding theorems on domains and fractal sets in ${\bf R}^n$.

Poincar\'e inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces