The main theme of this thesis is the formulation of self-duality for extended objects (p-branes). An approach to self-duality for membranes is developed using the correspondence between the large N limit of SU(N) gauge theories and the membrane theory. This correspondence is established via the use of the coadjoint orbit method. It is shown that classical gauge field theories can be formulated on the coadjoint orbits of an infinite dimensional group (a semidirect product of the group of gauge transformations and the Heisenberg-Weyl group); in Chap. II this construction is carried out for Yang-Mills, Cherns-Simons, topological Yang-Mills and F A B theories, as well as the Wess-Zumino-Novikov-Witten model. In Chap. III it is shown that for homogeneous fields (i.e. gauge mechanics) and in the N -1-" limit, the coadjoint orbit action becomes identical to the membrane action in the light cone gauge. The self-duality equations for gauge fields then translate into the self-duality equations for membranes.

In Chap. IV another approach is developed, one which allows us to formulate the self-duality equations for a much larger class of extended objects. This generalized self-duality is based on the notion of p-fold vector products. We exhibit several classes of solutions for these generalized self-dual extended objects and classify all the cases in which they exist. We also show that the self-intersecting string instantons, introduced by Polyakov constitute a special case of these solutions. Of particular interest are two octonionic classes: a membrane in 7 dimensions and a 3-brane in 8 dimensions. To simplify the calculations in these cases we developed an approach to octonionic symbolic computing making use of "Mathematica". Some possible applications of self-dual extended objects are briefly discussed.