[en] In this thesis we present research related to the general areas of geometry and supersymmetry as well as D-brane worldvolume soliton solutions. We investigate certain topological aspects of Kahler and hyperKahler manifolds from a supersymmetric point of view, using the interplay between geometry and supersymmetry in sigma models. We then analyze BPS worldvolume solitons for certain D-p-branes. Following an idea of Witten's, we first investigate how the cohomology of a compact (hyper-)Kahler manifold admits the action of the Lie algebra so(2,1) (resp.so(4,1)). These actions follow from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry. We go on to invstigate and prove the correspondence between certain octonionic instantons and triholomorphic curves. This generalizes the well-known correspondence in four-dimensional gauge theory between instantons and holomorphic curves. We prove that the former correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. Following this we turn to the investigation of BPS saturated worldvolume solitons in the D-3- and D-4-brane. For the D-3-brane we show how the central charges of the worldvolume supersymmetry algebra give a lower bound on the D-3-brane tension. We also show that a D-3-brane in a supergravity D-3-brane background admits finite energy, non-singular abelian monopoles and dyons preserving 1/2 or 1/4 of the N = 4 supersymmetry which saturate a BPS-type bound. We show that the 1/4 supersymmetric solitons provide a worldvolume realisation of string-junctions. We also include a brief discussion of the Hamiltonian formalism for the D-8-brane. (author)