This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in two and three dimensions in the quasistatic regime. Two key figures of CALR are (i) the localized resonance and (ii) the connection between the localized resonance and the blow up of the power of the fields as the loss 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 and analyzed in [8-10]. It was shown in [12] that the complementary property of media is not enough to ensure such a connection. In this paper, we introduce a subclass of complementary media called the class of doubly complementary media. This class is rich enough to allow us to do cloaking via anomalous localized resonance for an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 located in an arbitrary medium. The following three properties are established: 1) CALR appears if and only if the power blows up; 2) the power blows up if the source is "located" near the plasmonic structure; 3) 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. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.