The Pfaffian of a skew symmetric matrix of even order is the square root of its determinant. \textit{D. E. Knuth} [Electron. J. Comb. 3, No. 2, Research paper R5, 13 p. (1996); printed version J. Comb. 3, No. 2, 147-159 (1996; Zbl 0862.15007)] suggested a combinatorial approach to Pfaffians based on arguments of graph theory and obtained an elegant proof of a classical Pfaffian identity. In the paper under review the author combines the approach of Knuth with other combinatorial arguments. As a result, several known Pfaffian identities arising from physics are proved from a unique point of view, see e.g. \textit{R. Hirota} [Soliton solutions to the BKP equations. I. The Pfaffian technique, J. Phys. Soc. Japan 58, 2285-2296 (1989)], \textit{S. Tsujimoto} and \textit{R. Hirota} [J. Phys. Soc. Japan 65, No. 9, 2797-2806 (1996; Zbl 0948.39007)]. Another application is a Laplace expansion of the Pfaffians and a new proof of a result of \textit{H. Srinivasan} [J. Algebra 163, No. 2, 312-334 (1994; Zbl 0795.15004)]. The author also establishes a Pfaffian identity which appears to be new. The results in the paper have in common that they are all vector-based, in the sense that when summing or selecting entries of a matrix, the author considers vectors of elements rather than individual elements. Finally, the paper contains an application to symmetric functions proving an identity for Schur \(Q\)-functions.