Spurred by the new examples found by Kornel Szlachányi of a form of lax monoidal category, the author felt the time ripe to publish a reworking of Eilenberg and Kelly's original paper on closed categories appropriate to the laxer context. The new examples are connected with bialgebroids. In a separate paper with Stephen Lack, we have also used the concept to give an alternative definition of quantum category and quantum groupoid. Szlachányi has called the lax notion skew monoidal. This paper defines skew-closed category, proves Yoneda lemmas for categories enriched over such, and looks at closed cocompletion.