This paper studies the consensusability problem of discrete-time linear multi-agent systems over analog fading networks. It aims to decide whether there exists a distributed controller such that the underlying multi-agent system can achieve mean square consensus over analog fading channels. Conditions to ensure mean square consensus are derived for the scenarios of undirected communication topologies with identical fading networks, balanced directed communication topologies with identical fading networks, and undirected communications topologies with non-identical fading networks, respectively. For scalar systems, the sufficient condition is shown to be necessary. The results indicate that the effect of fading networks on consensusability is determined by the statistics of channel fading. Finally, simulations are conducted to validate the theoretical results. (C) 2016 Elsevier Ltd. All rights reserved.