With the advent of cloud computing, organizations are nowadays able to react rapidly to changing demands for computational resources. Not only individual applications can be hosted on virtual cloud infrastructures, but also complete business processes. This allows the realization of so-called elastic processes, i.e., processes which are carried out using elastic cloud resources. Despite the manifold benefits of elastic processes, there is still a lack of solutions supporting them.In this paper, we identify the state of the art of elastic Business Process Management with a focus on infrastructural challenges. We conceptualize an architecture for an elastic Business Process Management System and discuss existing work on scheduling, resource allocation, monitoring, decentralized coordination, and state management for elastic processes. Furthermore, we present two representative elastic Business Process Management Systems which are intended to counter these challenges. Based on our findings, we identify open issues and outline possible research directions for the realization of elastic processes and elastic Business Process Management.