One goal of multiobjective optimization is to approximate the nondominated set in the image space. One widely used approximation concept is that of enclosures. These are unions of closed boxes that cover the nondominated set. The bounds of these boxes form the lower and upper bound sets of the enclosure. The quality of an enclosure is measured by its width. For the bounding sets, a frequently used concept is that of local lower and upper bound sets. The upper and lower search regions are derived from finite and stable point sets that lie within the interior of an initial box. The upper and lower search zones induced by the finite local lower and upper bound sets represent the upper and lower search regions. If the nondominated set lies within the interior of the initial box and is not dominated in some sense by the points of the point sets, then the local lower and upper bound sets provide an approximation of the nondominated set. In this chapter, we provide an introduction to enclosures and local lower and upper bound sets based on the current literature in these fields.