Building on the work of Grothendieck on tensor products and Fredholm determinants, the authors develop a theory of relative Pfaffians for operators (resp. bilinear forms) on Banach spaces. In the finite dimensional case, the relative Pfaffian of two skew-symmetric \(2k\times 2k\) matrices \(A\) and \(B\) (\(A\) being invertible) is defined to be \[ \text{Pf}(A,B)={{\text{Pf}(A^{-1}-B)} \over {\text{Pf}(A^{-1})}}, \] where \(\text{Pf}(A)\) is the Pfaffian of \(A\). In the infinite dimensional one it is defined by the infinite series \[ \text{Pf}(A,B)=\sum_{n=0}^ \infty {1 \over (n!)^ 2}\langle \bigwedge^ n A,\bigwedge^ n B\rangle, \] where \(A\) and \(B\) are alternating bilinear forms (\(A\) on \(E'\) and \(B\) on \(E\)). The paper is devoted to the derivation of natural algebraic identities for the Pfaffian resp. to the study of the relative Pfaffian minors which play a role in the theory of Pfaffians corresponding to that of the Fredholm minors in that of the Fredholm determinant. (The phrase ``Since \(E'\otimes E\) is naturally identified with the space of nuclear operators on \(E\)'' suggests that the authors are tacitly assuming that all Banach spaces which occur have the approximation property).