In practical implementations, meshless methods using moving least-squares (MLS) approximation and global Galerkin formulation require a background mesh for domain integration. An adaptive procedure based on background mesh is developed for meshless methods using MLS. It comprises a cell energy error estimate and a local domain refinement technique. The error estimate differs from conventional pointwise approaches in that it evaluates error based on individual cells instead of points. For each cell, a computed cell energy and a reference cell energy are generated based on a same stress field by using two different integration schemes; the difference between the two energy values is used as the basic measure of error estimation. The advantage of this strategy is that it requires only one stress field and no reference field needs to be furnished. This has significance, as the stress field generated by meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for finite element methods are not applicable. To achieve high efficiency in domain refinement, a local technique based on the Delaunay algorithm is implemented. In this technique, each node is assigned a scaling factor to control local nodal density; refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor. The numerical experiments in this paper show that the proposed adaptive procedure is simple, effective and efficient. The limitations of the approach are also discussed.