Some general results on uniform convergence and saturation of positive operators are proved. The smoothness of the functions is measured by \(\omega_{\phi}(f,\delta)=\sup_{0\leq h\leq \delta_ ix}| \Delta^ 2_{h\phi (x)}f(x)|\) which allows one to obtain some estimates. Under very general conditions on the positive operators \(\{L_ n\}\) one has e.g. \(| L_ nf-f| \leq K\omega_{\phi}(f,\alpha_ n)\) where \(\phi\) and \(\alpha_ n\) are related with the second moments of \(f: L_ n((-x)^ 2;x)\leq K_ 1\phi^ 2(x)\alpha^ 2_ n.\) As an illustration we quote: if \(S_ n(f,x)=\sum^{\infty}_{k=0}f(k/n)e^{-nx}(nx)^ k/k!\) are the Szász-Mirakjan operators then \(S_ nf\to f\) uniformly on [0,\(\infty)\) if and only if \(f(x^ 2)\) is uniformly continuous on [0,\(\infty)\).