  The goal of this work is to generalize the Gauss-Bonnet and Poincar\'{e}-Hopf
Theorems to the case of orbifolds with boundary. We present two such
generalizations, the first in the spirit of Satake. In this case, the local
data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem
and the index of the vector field in the case of the Poincar\'{e}-Hopf Theorem)
is related to Satake's orbifold Euler characteristic, a rational number which
depends on the orbifold structure. For the second pair of generalizations, we
use the Chen-Ruan orbifold cohomology to express the local data in a way which
can be related to the Euler characteristic of the underlying space of the
orbifold. This case applies only to orbifolds which admit almost-complex
structures.
