Astronomical transients are stellar objects that become temporarily brighter on various timescales and have led to some of the most significant discoveries in cosmology and astronomy. Some of these transients are the explosive deaths of stars known as supernovae while others are rare, exotic, or entirely new kinds of exciting stellar explosions. New astronomical sky surveys are observing unprecedented numbers of multi-wavelength transients, making standard approaches of visually identifying new and interesting transients infeasible. To meet this demand, we present two novel methods of quickly and 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 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 show that the flexibility of neural networks, the attribute that makes them such a powerful regressor for many tasks, is what makes them less suitable for anomaly detection when compared with our parametric model.