The vigorous development of positioning technology and pervasive computing has given rise to trajectory big data, i.e. the high speed trajectory data stream that originated from positioning devices. Analyzing trajectory big data timely and effectively enables us to discover the abnormal patterns that hide in trajectory data streams, and therefore to provide effective support to applications such as urban planning, traffic management, and security controlling. The traditional anomaly detection algorithms cannot be applied to outlier detection in trajectory big data directly due to the characteristics of trajectories such as uncertainty, un-limitedness, time-varying evolvability, sparsity and skewness distribution. In addition, most of trajectory outlier detection methods designed for static trajectory dataset usually assume a priori known data distribution while disregarding the temporal property of trajectory data, and thus are unsuitable for identifying the evolutionary trajectory outlier. When dealing with huge amount of low-quality trajectory big data, a series of issues need to be addressed. Those issues include coping with the concept drifts of time-varying data distribution in limited system resources, online detecting trajectory outliers, analyzing causal interactions among traffic outliers, identifying the evolutionary related trajectory outlier in larger spatial-temporal regions, and analyzing the hidden abnormal events and the root cause in trajectory anomalies by using application related multi-source heterogeneous data. Aiming at solving the problems mentioned above, this paper reviews the existing trajectory outlier detecting techniques from several categories, describes the system architecture of outlier detection in trajectory big data, and discusses the research directions such as outlier detection in trajectory stream, visualization and evolutionary analysis in trajectory outlier detection, benchmark for trajectory outlier detection system, and data fusion in semantic analysis for anomaly detection results.