Time-domain astronomy has reached an incredible new era where unprecedented amounts of data are becoming available with new surveys such as LSST expected to observe over 10 million transient alerts every night. In this talk, I'll discuss the issue that with such large data volumes, the astronomical community will struggle to identify rare and interesting anomalous transients that have previously been found serendipitously. I'll present two methods of automatically detecting anomalous transient light curves in real time. The first modelling approach is a probabilistic neural network built using Temporal Convolutional Networks (TCNs) and the second is an interpretable Bayesian parametric model of a transient. We demonstrate our methods' ability to provide anomaly scores as a function of time on light curves from the Zwicky Transient Facility. We show that the flexibility of neural networks, the attribute that makes them such a powerful tool for many regression tasks, is what makes them less suitable for anomaly detection when compared with our parametric model. The parametric model is able to identify anomalies with respect to common supernova classes with high precision and recall for most rare classes such as kilonovae and tidal disruption events. Our ability to identify anomalies improves over the lifetime of the light curves. Our framework, used in conjunction with transient classifiers, will enable fast and prioritised follow-up of unusual transients from new surveys.