This paper presents a novel application of the theory of conformal prediction for distribution-independent on-line learning and anomaly detection. We exploit the fact that conformal predictors give valid prediction sets at specified confidence levels under the relatively weak assumption that the (normal) training data together with (normal) observations to be predicted have been generated from the same distribution. If the actual observation is not included in the possibly empty prediction set, it is classified as anomalous at the corresponding significance level. Interpreting the significance level as an upper bound of the probability that a normal observation is mistakenly classified as anomalous, we can conveniently adjust the sensitivity to anomalies while controlling the rate of false alarms without having to find any application specific thresholds. The proposed method has been evaluated in the domain of sea surveillance using recorded data assumed to be normal. The validity of the prediction sets is justified by the empirical error rate which is just below the significance level. In addition, experiments with simulated anomalous data indicate that anomaly detection sensitivity is superior to that of two previously proposed methods.