Aircraft trajectory prediction supports trajectory-based operations and air traffic management by estimating future aircraft motion. This systematic review synthesizes studies on trajectory, position, and flight state prediction for civil, manned, fixed-wing aircraft during airborne operation from takeoff to touchdown. Following evidence assessment, 198 studies were retained as primary evidence, of which 55 met the predefined criteria for formal performance synthesis. The review uses a Preparation, Prediction, Update, and Output (PPUO) framework and organizes the evidence around modeling methodology, task formulation, and performance evaluation. The findings show that physics-based, state estimation, data-driven, and hybrid methods have different modeling assumptions and operating conditions, and the available evidence does not support any one paradigm as universally superior across tasks and operating scenarios. Reported performance is jointly influenced by prediction horizon, target cardinality, interaction context, and evaluation conditions. The evidence is concentrated on single-target prediction; quantitative evidence for joint multi-aircraft prediction and probabilistic uncertainty evaluation remains limited, while heterogeneity in experimental conditions restricts direct cross-study comparison. Future research should establish reproducible benchmarks with complete reporting of experimental conditions and conduct cross-scenario validation while evaluating trajectory accuracy, uncertainty quality, computational efficiency, and operational assurance together.