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We start by introducing the concept of so-called weak solutions to partial differential equations and of Sobolev spaces, which play an important role in the modern mathematical analysis. We then explore general ideas behind so-called Sobolev embeddings and their compactness. After the introduction, we will focus on quantitative analysis of noncompactness of Sobolev embeddings. Some selected recent results from the speaker’s research will be presented. At the end, we briefly discuss Sobolev trace embeddings and their compactness in the general setting of so-called rearrangement-invariant function spaces.