Function spaces of logarithmic smoothness : embeddings and characterizations In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov Sobolev Triebel Lizorkin). We establish the following results: (1) Sharp embeddings between the Besov spaces defined by differences and by Fourier analytical decompositions as well as between Besov and Sobolev Triebel Lizorkin spaces; (2) Various new characterizations for Besov norms in terms of different K functionals. For instance we derive characterizations via ball averages approximation methods heat kernels and Bianchini type norms; (3) Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.