This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two- dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance and the blow up of the power of the fields as the loss of the material goes to 0. An important class of negative index materials for which the localized resonance might appear is the class of reflecting complementary media introduced in [24]. It was shown in [29] that the complementarity property is not enough to ensure a connection between the blow up of the power and the localized resonance. In this paper, we study CALR for a subclass of complementary media called doubly complementary media. This class is rich enough to allow us to cloak an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 placed in an arbitrary medium via anomalous localized resonance; the cloak is independent of the source. The following three properties are established for doubly complementary media: P1. CALR appears if and only if the power blows up; P2. The power blows up if the source is located "near" the plasmonic structure; P3. The power remains bounded if the source is far away from the plasmonic structure. Property P2, the blow up of the power, is in fact established for reflecting complementary media. The proofs are based on several new observations and ideas. One of the difficulties is to handle the localized resonance. To this end, we extend the reflecting and removing localized singularity techniques introduced in [24-26], and implement the separation of variables for Cauchy problems for a general shell. The results in this paper are inspired by and imply recent ones of Ammari et al. [3] and Kohn et al. [16] in two dimensions and extend theirs to general non-radial core-shell structures in both two and three dimensions.