Abstruct: Embeddings of a discrete metric space into a Hilbert spaces or a "good'' Banach space have found many significant applications. At the beginning of the talk I plan to give a brief description of such applications. After that I plan to present three of my results: (1) On L1-embeddability of graphs with large girth; (2) Embeddability of infinite locally finite metric spaces into Banach spaces is finitely determined; (3) Metric characterization of the Radon-Nikodym property.