\begin{abstract} An {\em ornament} is a collection of oriented closed curves in a plane or another 2-surface, none three of which intersect at the same point (see \cite{V}). Similarly a {\em doodle} is a collection of oriented closed curves with no triple points at all. Homotopy invariants of ornaments and doodles are natural analogues of homotopy and isotopy invariants of links respectively. The Vassiliev theory of {\em finite-order invariants} of ornaments and the series of such invariants constructed in \cite{Mfb} is applied to doodles. It appears that these finite-order invariants classify doodles. Similar finite-order invariants of connected oriented closed plane curves classify the latter up to isotopy of the ambient plane. \end{abstract} \bibitem[M98]{Mfb} A.B.Merkov, {\em Vassiliev invariants classify flat braids}, preprint 1996, to appear in Differential and simplectic topology of knots and curves, ser. Progress in Math. Sciences, S.L.Tabachnikov (ed.) AMS, Providence RI, 1998. \bibitem[V93]{V} V.A.Vassiliev, {\em Invariants of ornaments}, Preprint Maryland Univ., March 1993; {\em Singularities and Bifurcations}, AMS, Advances in Soviet Math., vol.~21 (ed. V.I.Arnold), Providence, R.I., 1994, p. 225--261.