  This thesis is devoted to the study of well-posedness properties of some
geometric variational problems: existence, regularity and uniqueness of
solutions. We study two specific problems arising in the context of geometric
calculus of variations and sharing strong analogies: the Plateau's problem and
the optimal branched transport problem. The first part of the thesis discusses
the existence theory. Both problems are formulated in the language of Federer
and Fleming's theory of currents. After an exposition of the main results, we
will present the core ideas of the (interior) regularity theory for
area-minimizing currents and for optimal transport paths. The last part of the
thesis contains two original results: the generic uniqueness of solutions both
for the Plateau's problem (in any dimension and codimension) and for the
optimal branched transport problem.
