Cloud computing represents the most recent enterprise trend in information technology and refers to the virtualization of computing resources that are available on demand. Cloud computing saves cost and time for businesses. Moreover, this computing process reflects a radical technological revolution in how companies develop, deploy, and manage enterprise applications over the Internet. Virtualized cloud computing mainly offers cloud-computing delivery models such as software as a service, platform as a service, and infrastructure as a service. Security and privacy are presently considered critical factors in the adaptation of any cloud-service delivery model. Cloud computing leverages several technologies; in the process, this model can inherit potential security threats. Thus far, security issues in cloud computing have rarely been addressed at the service delivery level. Key security concerns include Web application security, network security, data security, integration, vulnerabilities in the virtualized environment, and physical security. The aim of this research is to comprehensively present the security threats with respect to their cloud service deliver models. This study also determines how service delivery models differ from existing enterprise applications, classify these models, and investigate the inherent security challenges. This study primarily focuses on the security concerns of each layer of the cloud-service delivery model, as well as on existing solutions and approaches. In addition, countermeasures to potential security threats are also presented for each cloud model. Copyright (C) 2016 John Wiley & Sons, Ltd.