The standard Artin approximation (AP) is not readily applicable to the left-right equivalence of map-germs Maps((k^n,o),(k^m,o)). Moreover, the naive extension does not hold in the k-analytic case, because of Osgood-Gabrielov-Shiota examples. The left-right version of Artin approximation (LRAP) was established by M. Shiota for map-germs that are either Nash or [real-analytic and of finite singularity type]. We establish LRAP for Maps(X,Y) where X,Y are k-analytic/k-Nash germs of schemes of any characteristic. More precisely: * (k-Nash power series, k<x>/J_X and k<y>/J_Y.) LRAP holds for any map. * (k-analytic power series, k{x}/J_X and k{y}/J_Y.) LRAP holds for maps of weakly-finite singularity type. This ``weakly-finite singularity type" (which we introduce) is a natural extension of the classical finite singularity type, and is of separate interest. As a trivial corollary we get: the inverse Artin approximation holds for analytic maps of weakly-finite singularity type. Then we extend the properties RAP, LAP, LRAP to the approximation results for quivers of maps, Gamma-AP. Finally, we establish the nested version, ``Gamma-AP with parameters". It is needed for families/unfoldings of maps.