We propose an approach to detecting anomalies and providing basic diagnostics of these anomalies from aircraft cruise flight data. The detection and diagnostics are based on a model learned from the historical data of a fleet of aircraft. For a variety of cruise flight conditions with and without turbulence, we validate the approach using a simulated FOQA dataset generated by a NASA flight simulator. We identify a regression model that maps the flight conditions and aircraft control inputs into accelerations (linear and rotational). Anomalies are detected as outliers that exceed the scatter caused by turbulence and the modeling error. The detection method is related to multivariable statistical process control. The data-driven model represents aircraft dynamics and establishes signatures of the faults in aircraft actuators and sensors. Using multiple hypothesis testing, we are able to diagnose the fault condition that caused the anomaly. The diagnosis is not perfect; instead, we winnow the faults down to an ambiguity group. Offsets that are a fraction of a degree in flight surface position and a small percentage of the experienced sensor range are reliably detected from the data collected under light turbulence conditions.