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

The Poincar\'e Lemma in Subriemannian Geometry