Providing a lightweight self-adaptive approach that does not need offline training in advance and meanwhile is able to detect anomalies in real time could be highly beneficial. Such an approach could be immediately applied and deployed on any commodity machine to provide timely anomaly alerts. To facilitate such an approach, this paper introduces SALAD, which is a Self-Adaptive Lightweight Anomaly Detection approach based on a special type of recurrent neural networks called Long Short-Term Memory (LSTM). Instead of using offline training, SALAD converts a target time series into a series of average absolute relative error (AARE) values on the fly and predicts an AARE value for every upcoming data point based on short-term historical AARE values. If the difference between a calculated AARE value and its corresponding forecast AARE value is higher than a self-adaptive detection threshold, the corresponding data point is considered anomalous. Otherwise, the data point is considered normal. Experiments based on a real-world time series dataset demonstrates that SALAD outperforms five other state-of-the-art anomaly detection approaches in terms of detection accuracy. In addition, the results also show that SALAD is lightweight and can be deployed on a commodity machine.