Summary: We give a clear, fairly self-contained account of the problem of the center and the question of integrability for plane quadratic vector fiels. The literature on this subject contains many inaccuracies and errors and there exists no survey covering all aspects of the problem and warning the reader about these pitfalls. This article is designed to fill this gap. However, we also give a new proof for the sufficiency of the conditions for the center, a proof which simultaneously shows the “global” integrability of the systems satisfying these conditions. Their first integrals are elementary functions, defined on \({\mathbb{R}}^ 2\) or the complement of some algebraic curve.