New time-domain surveys, such as the Rubin Observatory Legacy Survey of Space and Time (LSST), will observe millions of transient alerts each night, making standard approaches of visually identifying new and interesting transients infeasible. We present two novel methods of automatically detecting anomalous transient light curves in real-time. Both methods are based on the simple idea that if the light curves from a known population of transients can be accurately modelled, any deviations from model predictions are likely anomalies. 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 low false anomaly rates and high true anomaly rates achieving Area Under the Receive Operating Characteristic (ROC) Curve (AUC) scores above 0.8 for most rare classes such as kilonovae, tidal disruption events, intermediate luminosity transients, and pair-instability supernovae. 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 large-scale surveys.