Grid computing provides mechanisms for making large-scale computing environments available to the masses. In recent times, with the advent of Cloud computing, the concepts of Software as a Service (SaaS), where vendors provide key software products as services over the internet that can be accessed by users to perform complex tasks, and Service as Software (SaS), where customizable and repeatable services are packaged as software products that dynamically meet the demands of individual users, have become increasingly popular. Both SaaS and SaS models are highly applicable to scientific software and users alike. Opal2 is a toolkit for wrapping scientific applications as Web services on Grid and cloud computing resources. It provides a mechanism for scientific application developers to expose the functionality of their codes via simple Web service APIs, abstracting out the details of the back-end infrastructure. Services may be combined via customized workflows for specific research areas and distributed as virtual machine images. In this paper, we describe the overall philosophy and architecture of the Opal2 framework, including its new plug-in architecture and data handling capabilities. We analyze its performance in typical cluster and Grid settings, and in a cloud computing environment within virtual machines, using Amazon's Elastic Computing Cloud (EC2).