  This work is a short, self-contained introduction to subriemannian geometry
with special emphasis on Chow's Theorem. As an application, a regularity result
for the Poincar\'e Lemma is presented.
  At the beginning, the definitions of a subriemannian geometry, horizontal
vector fields and horizontal curves are given. Then the question arises: Can
any two points be connected by a horizontal curve? Chow's Theorem gives an
affirmative answer for bracket generating distributions. (A distribution is
called bracket generating if horizontal vector fields and their iterated Lie
brackets span the whole tangent space.)
  We present three different proofs of Chow's Theorem; each one is interesting
in its own. The first proof is based on the theory of Stefan and Sussmann
regarding integrability of singular distributions. The second proof is
elementary and gives some insight in the shape of subriemannian balls. The
third proof is based on infinite dimensional analysis of the endpoint map.
  Finally, the study of the endpoint map allows us to prove a regularity result
for the Poincar\'e Lemma in a form suited to subriemannian geometry: If for
some $r \geq 0$ all horizontal derivatives of a given function $f$ are known to
be $r$ times continuously differentiable, then so is $f$.
  Sections 1 to 3 are the common work of Martin Bauer and Philipp Harms.
