Diagrammatic Pairing Fluctuations Approach to the BCS-BEC Crossover

This paper gives a survey of a diagrammatic approach for fermionic pairing fluctuations, which are relevant to the BCS-BEC crossover real-ized with ultracold Fermi gases. Emphasis will be given to the physical intuition about the relevant physical processes that can be associated with this approach. Specific results will be presented for thermody-namic and dynamical quantities, where a critical comparison with al-ternative diagrammatic approaches will also be attempted. [To be published as a chapter in BCS-BEC Crossover and the Uni