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.