Summary: We study the degree of approximation by superpositions of a sigmoidal function. We mainly consider the univariate case. If \(f\) is a continuous function, we prove that for any bounded sigmoidal function \(\sigma\), \(d_{n,\sigma}(f)\leq \|\sigma\| \omega \bigl( f,{1\over{n+1}}\bigr)\). For the Heaviside function \(H(x)\), we prove that \(d_{n,H}(f)\leq \omega \bigl( f, {1\over {2n+1)}} \bigr)\). If \(f\) is a continuous function of bounded variation, we prove that \(d_{n,\sigma} (f)\leq {{\|\sigma\|} \over {n+1}} V(f)\) and \(d_{n,H}\leq {1\over {2n+1}} V(f)\). For the Heaviside function, the coefficient 1 and the approximation orders are the best possible. We compare these results with the classical Jackson and Bernstein theorems, and make some conjectures for further study.