This review summarizes the features of cold shock domain (CSD) proteins in the context of their interactions with nucleic acids and describes similarities and differences in the structure of cold shock proteins of prokaryotes and CSD proteins of eukaryotes with special emphasis on the functions related to the RNA/DNA-binding ability of these proteins. The mechanisms and specificity of their interaction with nucleic acids in relation to the growing complexity of protein domain structure are described, as well as various complexes of the mammalian Y-box binding protein 1 (YB-1) with nucleic acids (filaments, globules, toroids). The role of particular amino acid residues in the binding of nitrogenous bases and the sugar-phosphate backbone of nucleic acids is emphasized. The data on the nucleic acid sequences recognized by the Y-box binding proteins are systematized. Post-translational modifications of YB-1, especially its phosphorylation, affect the recognition of specific sequences in the promoter regions of various groups of genes by YB-1 protein. The data on the interaction of Lin28 protein with let-7 miRNAs are summarized. The features of the domain structure of plant CSD proteins and their effect on the interaction with nucleic acids are discussed.