The author announces an algorithm for determining whether a given element of the fundamental group \(\pi_1(M)\) of a connected compact 2-manifold \(M\) contains representatives which are simple closed curves. The algorithm is stated for the case \(M\) orientable, with or without boundary. The method is based on the concept of the winding number of a regular closed curve \(\gamma\) with respect to a vector field \(X\) on \(M\) which does not vanish anywhere on \(\gamma\). Roughly speaking, the winding number is the total increase in the angle between the tangent to \(\gamma\) at \(x\) and the vector \(X(x)\) as \(x\) travels round \(\gamma\), divided by \(2\pi\). Winding numbers on surfaces were introduced and studied by \textit{B. L. Reinhart} [Ann. Inst. Fourier 10, 271--283 (1960; Zbl 0097.16203)]. By choosing suitable regular representatives of each element of \(\pi_1(M)\) the author here defines a function \(\omega_X\) (also called winding number) from \(\pi_1(M)\) to the integers when \(M\) is orientable with non-empty boundary and \(X\) is a non-vanishing vector field on \(M\). This \(\omega_X\) is not a homomorphism: the way in which it differs from a homomorphism is used to provide the algorithm for simple curves. A given element \(c\in \pi_1(M)\) can be written (not uniquely) as a word \(w\) in standard generators (and their inverses) of \(\pi_1(M)\). A method of reduction due to \textit{H. Zieschang} [Math. Scand. 17(1965), 17--40 (1966; Zbl 0151.33005)] is applied to \(w\), producing a new word \(W\). It is stated, and proved elsewhere that unless \(W\) consists of only one symbol or is of the form \(t^k\) \((k\geq 2)\) then the non-trivial element \(c\) fails to contain simple curves if and only if some cyclic permutation of \(W\) is expressible as \(uv\) with \(\omega_X(uv^{-1})\neq \omega_X(u) +\omega_X(v^{-1})\). A simple formula is given for computing \(\omega_X\) of a word. The algorithm then consists of the calculation of \(\omega_X(uv^{-1}) - \omega_X(u) - \omega_X(v^{-1})\) (which is independent of \(X\)) for at most all the possible `decompositions' \(uv\) of cyclic permutations of \(W\). The extension to cover the case \(\partial M=\emptyset\) is easy. A similar algorithm applies when \(M\) is non-orientable, using winding numbers in \(\mathbb Z_2\). The definition of \(\omega_X\) and some applications of winding numbers, apart from the derivation of the above algorithm, are to appear elsewhere.