Abstract: About a century ago Burnside raised the idea of finiteness in group theory by asking whether a finitely generated (f.g.) periodic group is necessarily finite.  This question gave birth to the study of finiteness conditions in algebra and led to the proliferation of Burnside type problems.  Probably the most striking among these problems is the 1941 question of Kurosh about the existence of a f.g. infinite dimensional algebraic algebra.  This problem is in fact strongly related to that of Burnside. Actually, in 1945 Jacobson observed that a counterexample to the Kurosh problem gives a counterexample to the Burnside problem.  However, Kurosh's question is also connected to other difficult questions such as the Koethe and Amitsur conjectures.  Although there are many counterexamples to the Burnside problem, for a long time, however the only known counterexamples to both the Burnside and Kurosh problems were those constructed by Golod, using the well known Golod-Shafarevich lemma.  Recently Smoktunowicz constructed a 3 generated nil algebra over a denumerable field such that its algebra of polynomials with two commuting indeterminates is not nil.  These algebras cannot be constructed over a nondenumerable field, since every f.g. nil algebra over a nondenumerable field is absolutely nil.  Our main goal here is to develop a technique to give new counterexamples to the Burnside and Kurosh problems. We mainly show that for any integer $d\geq 2$ and over any field ${\mathbb {K}}$, we construct a residually finite non nilpotent nil algebra over ${\mathbb {K}}$ generated by $d$ elements such that any $d-1$ elements generate a nilpotent subalgebra.  This leads to analogous results for nonassociative algebras and groups.  We shall point out that we made the effort to prove that our technique gives similar results to Golod's.  On the other hand, although our algebras and groups satisfy the Golod-Shafarevich lemma, the proof is completely different, as we do not use this lemma.  Our algebras are also different from Smoktunowicz's as we carry out our construction over every field, which leads to absolutely nil algebras.