let $w = \prod x^{a_i} y^{b_i}$ be an element of a free group on two generators $x, y$. for every group $g$ the corresponding word map $w : g^2 \to g$ is defined as $w(g_1, g_2) = \prod g_1^{a_1} g_2^{a_2}$. i will speak about surjectivity and equidistribution of such word maps on groups $\operatorname{psl}(2)$ and $\operatorname{sl}(2)$. this is a report on the joint works with sh. garion, f. grunewald, b.kunyavskii, eu. plotkin, and n. gordeev.