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We study the class of Bernstein algebras that are algebraic, in the sensethat each element generates a finite-dimensional subalgebra. Every Bernsteinalgebra has a maximal algebraic ideal, and the quotient algebra is azero-multiplication algebra. Several equivalent conditions for a Bernsteinalgebra to be algebraic are given. In particular, known characterizations oftrain Bernstein algebras in terms of nilpotency are generalized to the case oflocally train algebras. Along the way, we show that if a Banach Bernsteinalgebra is algebraic (respectively, locally train), then it is of boundeddegree (respectively, train).  Then we investigate the Kurosh problem for Bernstein algebras: whether afinitely generated Bernstein algebra which is algebraic of bounded degree isfinite-dimensional. This problem turns out to have a closed link with aquestion about associative algebras. In particular, when the barideal is nil,the Kurosh problem asks whether a finitely generated Bernstein-train algebra isfinite-dimensional. We prove that the answer is positive for some specificcases and for low degrees, and construct counter-examples in the general case.  By results of Yagzhev, the Jacobian conjecture is equivalent to a certainstatement about Engel and nilpotence identities of multioperator algebras. Weshow that the generalized Jacobian conjecture for quadratic mappings holds forBernstein algebras.

Bernstein algebras that are algebraic and the Kurosh problem