  Let $R$ be an associative algebra over a field $K$ generated by a vector
subspace $V$. The polynomial $f(x_1,\ldots,x_n)$ of the free associative
algebra $K\langle x_1,x_2,\ldots\rangle$ is a weak polynomial identity for the
pair $(R,V)$ if it vanishes in $R$ when evaluated on $V$. We survey results on
weak polynomial identities and on their applications to polynomial identities
and central polynomials of associative and close to them nonassociative
algebras and on the finite basis problem. We also present results on weak
polynomial identities of degree three.
