The ability to robustly detect abnormal driving behavior has the potential to limit traffic accidents and save many lives. Abnormal driving behavior that threatens road safety includes aggressive, anxious, nervous, and unstable driving. Any of these can lead to dangerous situations in traffic. Therefore, we aim to provide a robust mechanism to detect such abnormal driving behavior. In this paper, we present our work in this regard which focuses on neural networks-based anomaly detection approaches. We consider autoencoder replicator neural networks and long short-term memory networks; comparing them to a previously employed Isolation Forest. We show that introducing a post-processing approach, that takes into account the recent history of a vehicle, reliable anomaly detection for driving behavior can be achieved based on the recurrent neural network. Its performance is well suited for application in a large scale detection system for driver assistance or autonomous vehicles.