The pay-as-you-go model supported by existing cloud infrastructure providers is appealing to most application service providers to deliver their applications in the cloud. Within this context, elasticity of applications has become one of the most important features in cloud computing. This elasticity enables real-time acquisition/release of compute resources to meet application performance demands. In this thesis we investigate the problem of delivering cost-effective elasticity services for cloud applications. Traditionally, the application level elasticity addresses the question of how to scale applications up and down to meet their performance requirements, but does not adequately address issues relating to minimising the costs of using the service. With this current limitation in mind, we propose a scaling approach that makes use of cost-aware criteria to detect the bottlenecks within multi-tier cloud applications, and scale these applications only at bottleneck tiers to reduce the costs incurred by consuming cloud infrastructure resources. Our approach is generic for a wide class of multi-tier applications, and we demonstrate its effectiveness by studying the behaviour of an example electronic commerce site application. Furthermore, we consider the characteristics of the algorithm for implementing the business logic of cloud applications, and investigate the elasticity at the algorithm level: when dealing with large-scale data under resource and time constraints, the algorithm's output should be elastic with respect to the resource consumed. We propose a novel framework to guide the development of elastic algorithms that adapt to the available budget while guaranteeing the quality of output result, e.g. prediction accuracy for classification tasks, improves monotonically with the used budget.