The invention relates to the monitoring of pressure in production tubing (3) and/or annuli (7, 9) of an oil or gas well. Patterns of pressure in tubing and/or annuli may be indicative of a fault or a condition which requires attention. Patterns may also be recognized that, e.g. based in rates of change, may indicate that a fault may be going to occur in the future. Pressure differences between tubing and annuli or between different annuli may be monitored in the same way.