Electricity spot market prices are notoriously difficult to model, let alone predict, because of their extreme volatility. Such volatility is reflected in so-called price spikes that may increase the spot price by an order of magnitude in a matter of hours. Spot market price series are also subject to many other types of phenomena, such as periodicities at different scales, and to mean reversion. We introduce a model for electricity spot market prices that includes both spikes and mean reversion. The model is based on a jump-diffusion process that is superimposed on a mean-reverting Ornstein-Uhlenbeck model. Mean reversion takes place at several different time and price scales, so as to reproduce the observed behavior of spot market prices correctly. The parameters of the model are calibrated with the Nord Pool spot market hourly price series using a maximum likelihood approach. The simulated price series obtained in this way very closely follows the statistical characteristics of the real price series.