Organizations rely on data-driven methods to understand and anticipate how their operational processes unfold. Such insights support decision-making in settings where actions must be taken while processes are still ongoing. Predictive process monitoring addresses this need by producing predictions during the execution of running process instances. Many existing approaches show strong results but are mainly developed and evaluated in static, offline settings. These assumptions do not match online environments, where event data arrives continuously and processes change over time. Moving from an offline to an online setting introduces several challenges. Predictions are to be generated from incomplete, delayed, or irregular event streams. At the same time, models trained on historical data may lose validity as process behavior evolves. This creates the risk of relying on predictions that no longer reflect the current state of the process. To address these challenges, this thesis studies how predictive process monitoring systems can maintain trust in streaming events and adaptability in real-time event streams. We define trust as whether streaming event data accurately represent the underlying process. We develop streaming anomaly detection methods based on next-activity prediction. Observed events are evaluated during execution by comparing them with predicted probability distributions. To improve detection quality in practice, a semi-supervised extension incorporates limited expert feedback to update the anomaly classifier. Adaptability refers to whether predictive models can maintain performance as process conditions change. We propose measurements to capture changes in predictive performance over time. To cope with performance degradation, tailored retraining strategies are designed to improve prediction performance. Experiments on real-life and synthetic event logs show sustained performance with fewer unnecessary model updates. This thesis provides researchers and practitioners with approaches to assess trust in streaming event data and maintain adaptability of predictive models. It enables identifying unreliable data and sustaining reliable predictions under changing process conditions.