Statistical surveillance is used to detect an important change in a process as soon as possible after it has occurred, with control of false alarms. The EWMA, exponentially weighted moving average, method for surveillance is used in different areas, such as industry, economy and medicine. Three optimality criteria of surveillance are studied and the implications are described for the EWMA method and for suggested modifications. The first criterion concerns the average run length to alarm, ARL. This is the most commonly used criterion. Results on ARL optimality for EWMA are demonstrated. Equal weight for old and recent observations give good ARL-properties but bad properties otherwise. Thus, uncritical use of this criterion should be avoided. The second criterion is the ED criterion based on the minimal expected delay from change to detection. The full likelihood ratio method is optimal according to this criterion. Various approximations of this method tum out to be modifications of the EWMA method. Two of these modifications keep the EWMA statistic unchanged and just alter the alarm limits slightly. The approximations lead to a formula for the value of the weight parameter of the EWMA statistic. The usefulness of this formula is demonstrated. The conventional EWMA and the modifications are compared to the optimal full likelihood ratio method. No modification of EWMA is necessary for detection of large changes (where also the Shewhart 2 method is useful) but all the modifications give considerable improvement for small changes. The third criterion is based on the minimax of the expected delay after a change with respect to the time of the change. It is demonstrated that the value of the smoothing parameter, which is optimal according to this criterion, agrees well with that of the ED criterion but not with that of the ARL criterion. A restriction on the false alarm property is necessary. For the ARL criterion it is the ARL without any change. For the other two criteria we here use the false alarm probability. It is demonstrated that these two restrictions favor different methods.