We study a new challenging problem of efficient deployment for diverse tasks with different resources, where the resource constraint and task of interest corresponding to a group of classes are dynamically specified at testing time. Previous NAS approaches seek to design architectures for all classes simultaneously, which may not be optimal for some individual tasks. A straightforward solution is to search an architecture from scratch for each deployment scenario, which however is computation-intensive and impractical. To address this, we present a novel and general framework, called Elastic Architecture Search (EAS), permitting instant specializations at runtime for diverse tasks with various resource constraints. To this end, we first propose to effectively train the over-parameterized network via a task dropout strategy to disentangle the tasks during training. In this way, the resulting model is robust to the subsequent task dropping at inference time. Based on the well-trained over-parameterized network, we then propose an efficient architecture generator to obtain optimal architectures within a single forward pass. Experiments on two image classification datasets show that EAS is able to find more compact networks with better performance while remarkably being orders of magnitude faster than state-of-the-art NAS methods. For example, our proposed EAS finds compact architectures within 0.1 second for 50 deployment scenarios.