  In this Ph.D. thesis we review and elaborate on a method to find the D-brane
effective action, based on BPS equations. Firstly, both for the Yang-Mills
action and the Born-Infeld action it is shown that these configurations are
indeed BPS, i.e. solutions to these equations saturate a Bogomolny bound and
leave some supersymmetry unbroken. Next, we use the BPS equations as a tool to
construct the D-brane effective action and require that (a deformation of)
these equations should still imply the equations of motion in more general
cases. In the abelian case we managed to calculate all order in $\alpha'$
four-derivative corrections to the effective action and the BPS equations while
in the non-abelian case we obtained the effective action up to order
$\alpha'^4$. Furthermore, we discuss a check based on the spectrum of strings
stretching between intersecting branes. Finally, this Ph.D. thesis also
discusses the construction of a boundary superspace which is the first step to
use the method of Weyl invariance in N=2 superspace in order to again construct
the D-brane effective action. A more detailed summary of each chapter can be
found in the introduction.
