Let \(F: D\subseteq {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) be a continuously differentiable function such that F’(x) is regular for all \(x\in D\). To find a zero \(x^*\) of F in D, a number of good local methods are available. Such methods are globally convergent only under very restrictive conditions. The author shows that a suitable combination of any good local method with certain techniques from interval mathematics produces a strategy globally convergent under easily verifiable conditions. In particular, a quadratically convergent method is described. If the selected initial region contains no zero of F, this fact is detected after finitely many iterations. The use of rounded interval arithmetic provides rigorous (and small!) error bounds for the computed approximation to \(x^*\).