We study almost F -preenvelopes in the category of rings, for a significative class F of commutative rings. We completely identify those rings which have an almost F -preenvelope when F is the class of fields, semisimple rings, integer domains and local rings. We show that rings with Krull dimension zero have (almost) V-preenvelopes, where V is the class of von Neumann regular rings. key words. Almost preenvelopes, semiprimitive rings, local rings, von Neumann regular rings. AMS(MOS) subject classifications. 13C05; 16S10; 18B99.