In the Set Multicover problem, we are given a set system (X, S), where X is a finite ground set, and S is a collection of subsets of X. Each element x ∈ X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection S0 of S such that each point is covered by at least d(x) sets from S0. In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc. We give a polynomial time (2 + )-approximation algorithm for the set multicover problem (P, R), where P is a set of points with demands, and R is a set of non-piercing regions, as well as for the set multicover problem (D, P), where D is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands.