**Abstract** : In the present paper we prove some geometric properties of clothoids. A clothoid (or Cornu's spiral) is a plane curve whose curvature is a linear function of its length. In suitably chosen coordinates it is given by Frenel's integrals $x(t)=\int _0^t\cos (B\tau ^2/2)d\tau $, $y(t)=\int _0^t\sin (B\tau ^2/2)d\tau $. The clothoid appears in the problem to find (a) shortest plane curve(s) joining two given points with given at them tangent angles and curvatures and with a bounded derivative of the curvature. It turns out that a regular (i.e. of the class $C^3$) point of a shortest curve is either a piece of a clothoid or a line segment. Some of the properties exposed in the paper are used in \cite{K4} to show that when the distance between the initial and final points is sufficiently long, then, in general, shortest paths have infinitely many points of discontinuity of the curvature's derivative. Thus we are led to the problem to discribe a procedure of constructing paths (called »suboptimal») longer no more than a fixed constant than the shortest one(s) and which can be constructed explicitly. They are constructed from pieces of clothoids and a line segment (see \cite{K1}, \cite{K2}). The other properties of clothoids exposed in the paper are used in \cite{K1}, \cite{K2} to prove the suboptimal ity of the constructed paths.