This monograph studies an important generalisation of the Cauchy integral formula, called the Dbar formula. In particular, it investigates the physical significance of this formula, its extension to three, four and higher dimensions and the applications of the relevant formalism to: (i) The solution of boundary value problems for linear partial differential equations; (ii) The evaluation of real integrals; (iii) The construction of nonlinear integrable equations starting from the corresponding linear equations. A large part of this monograph is devoted to the theory of quaternions. It is shown that quaternions provide the proper generalisation of complex numbers. The book offers a pedagogical introduction to the theory of quaternions and attempts to elucidate its analytic component. Also, it presents some novel applications of this theory. This monograph will be useful to readers interested in the theory and applications of quaternions from the point of view of analysis as well as to everyone interested in explicit solutions of boundary value problems for partial differential equations.