  We develop a theory of $n \times n$-matrix Riemann-Hilbert problems for a
class of jump contours and jump matrices of low regularity. Our basic
assumption is that the contour $\Gamma$ is a finite union of simple closed
Carleson curves in the Riemann sphere. In particular, contours with cusps,
corners, and nontransversal intersections are allowed. We introduce a notion of
$L^p$-Riemann-Hilbert problem and establish basic uniqueness results and a
vanishing lemma. We also investigate the implications of Fredholmness for the
unique solvability and prove a theorem on contour deformation.
