Mobile malware and mobile network attacks are becoming a significant threat that accompanies the increasing popularity of smart phones and tablets. Thus in this paper we present our research vision that aims to develop a network-based security solution combining analytical modelling, simulation and learning, together with billing and control-plane data, to detect anomalies and attacks, and eliminate or mitigate their effects, as part of the EU FP7 NEMESYS project. These ideas are supplemented with a careful review of the state-of-the-art regarding anomaly detection techniques that mobile network operators may use to protect their infrastructure and secure users against malware.