Microservice-based systems (MSS) may fail with various fault types. While existing AIOps methods excel at detecting abnormal traces and locating the responsible service(s), human efforts are still required for diagnosing specific fault types and failure causes.This paper presents TraFaultDia, a novel AIOps framework to automatically classify abnormal traces into fault categories for MSS. We treat the classification process as a series of multi-class classification tasks, where each task represents an attempt to classify abnormal traces into specific fault categories for a MSS. TraFaultDia leverages meta-learning to train on several abnormal trace classification tasks with a few labeled instances from a MSS, enabling quick adaptation to new, unseen abnormal trace classification tasks with a few labeled instances across MSS. TraFaultDia's use cases are scalable depending on how fault categories are built from anomalies within MSS. We evaluated TraFaultDia on two MSS, TrainTicket and OnlineBoutique, with open datasets where each fault category is linked to faulty system components (service/pod) and a root cause. TraFaultDia automatically classifies abnormal traces into these fault categories, thus enabling the automatic identification of faulty system components and root causes without manual analysis. TraFaultDia achieves 93.26% and 85.20% accuracy on 50 new classification tasks for TrainTicket and OnlineBoutique, respectively, when trained within the same MSS with 10 labeled instances per category. In the cross-system context, when TraFaultDia is applied to a MSS different from the one it is trained on, TraFaultDia gets an average accuracy of 92.19% and 84.77% for the same set of 50 new, unseen abnormal trace classification tasks of the respective systems, also with 10 labeled instances provided for each fault category per task in each system.