Web application providers have been migrating their applications to cloud data centers, attracted by the emerging cloud computing paradigm. One of the appealing features of the cloud is elasticity. It allows cloud users to acquire or release computing resources on demand, which enables web application providers to automatically scale the resources provisioned to their applications without human intervention under a dynamic workload to minimize resource cost while satisfying Quality of Service (QoS) requirements. In this article, we comprehensively analyze the challenges that remain in auto-scaling web applications in clouds and review the developments in this field. We present a taxonomy of auto-scalers according to the identified challenges and key properties. We analyze the surveyed works and map them to the taxonomy to identify the weaknesses in this field. Moreover, based on the analysis, we propose new future directions that can be explored in this area.