The complexity of logic-based abduction has been extensively studied for the case in which the background knowledge is represented by a propositional theory, but very little is known about abduction with respect to description logic knowledge bases. The purpose of the current paper is to examine the complexity of logic-based abduction for the EL family of lightweight description logics. We consider several minimality criteria for explanations (set inclusion, cardinality, prioritization, and weight) and three decision problems: deciding whether an explanation exists, deciding whether a given hypothesis appears in some acceptable explanation, and deciding whether a given hypothesis belongs to every acceptable explanation. We determine the complexity of these tasks for general TBoxes and also for EL and EL+ terminologies. We also provide results concerning the complexity of computing abductive explanations.