| Summary: | In this thesis we construct closed expressions that correspond to the topological charges of the various BPS branes of the IIA, IIB and $D=11$ supergravity theories. These expressions are related to the structure of the SUSY algebras in curved spacetimes. We consider charges for all the M-, NS- and D-branes as well as the Kaluza-Klein monopoles. Additionally we consider the $SL(2,\mathbb{R})$ symmetry that exists in the IIB theory and $D=11$ theory in a double M9-brane background, and determine the charges for the remaining branes that fill up the $SL(2,\mathbb{R})$ multiplets. These include the charges corresponding to the multiplets of 7 and 9-branes in IIB as well as several new types of branes in $D=11$. We find that examining the possible multiplet structures of the charges provides another tool for exploring the spectrum of BPS branes that appear in these theories. Furthermore, we demonstrate how these charges map between theories. As a prerequisite to constructing some of the charges we determine the field equations and multiplet structure of the $D=11$ gauge potentials, extending previous results on the subject. The massive gauge transformations of the fields are also discussed, and we demonstrate how they are compatible with the construction of an $SL(2,\mathbb{R})$ covariant kinetic term in the $D=11$ Kaluza-Klein monopole worldvolume action.
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