The proliferation of services and service interactions within microservices and cloud-native applications, makes it harder to detect failures and to identify their possible root causes, which is, on the other hand crucial to promptly recover and fix applications. Various techniques have been proposed to promptly detect failures based on their symptoms, viz., observing anomalous behaviour in one or more application services, as well as to analyse logs or monitored performance of such services to determine the possible root causes for observed anomalies. The objective of this survey is to provide a structured overview and qualitative analysis of currently available techniques for anomaly detection and root cause analysis in modern multi-service applications. Some open challenges and research directions stemming out from the analysis are also discussed.