Let $A$ be an associative algebra over a field $F$ of characteristic zero, $F\langle X \rangle$ be the free algebra generated by the countable set $X=\{x_1,x_2,\ldots \}$ and $W$ be a unitary algebra over $F$. Then $A$ is called $W$-algebra if it has a structure of $W$-bimodule with some additional conditions. 

A generalized polynomial identity of $A$ is a polynomial $f(x_1,\ldots, x_n)$ of the free $W$-algebra $W\langle X\rangle$ that vanishes under all substitutions of the elements of $A.$ Roughly speaking, $f(x_1,\ldots, x_n)$ is a polynomial of $F\langle X \rangle$  with  ``coefficients" in $W.$ Notice that such ``coefficients" may appear also between two variables. Clearly these identities are a natural generalization of the ordinary polynomial ones arising when $W$ coincides with $F$. The set of all generalized polynomial identities $\mbox{GId}(A)$ is a $T_W$-ideal of $W\langle X\rangle,$ i.e., an ideal stable by endomorphisms of $W\langle X\rangle,$ and one of the main problems is to find a set of generators of such $T_W$-ideal. So far this problem has been achieved only for the algebra $M_n(F)$ of $n \times n$ full matrices over $F$ for all $n\geq 1$. 

The purpose of this talk is to present some recent results on the description of the generalized polynomial identities of another important and interesting algebra, the algebra $UT_2(F)$ of $2\times 2$ upper triangular matrices over $F$.