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With the growth of 5G, Internet of Things (IoT), edge computing and cloud computing technologies, the infrastructure (compute and network) available to emerging applications (AR/VR, autonomous driving, industry 4.0, etc.) has become quite complex. There are multiple tiers of computing (IoT devices, near edge, far edge, cloud, etc.) that are connected with different types of networking technologies (LAN, LTE, 5G, MAN, WAN, etc.). Deployment and management of applications in such an environment is quite challenging. In this paper, we propose ROMA, which performs resource orchestration for microservices-based 5G applications in a dynamic, heterogeneous, multi-tiered compute and network fabric. We assume that only application-level requirements are known, and the detailed requirements of the individual microservices in the application are not specified. As part of our solution, ROMA identifies and leverages the coupling relationship between compute and network usage for various microservices and solves an optimization problem in order to appropriately identify how each microservice should be deployed in the complex, multi-tiered compute and network fabric, so that the end-to-end application requirements are optimally met. We implemented two real-world 5G applications in video surveillance and intelligent transportation system (ITS) domains. Through extensive experiments, we show that ROMA is able to save up to 90%, 55% and 44% compute and up to 80%, 95% and 75% network bandwidth for the surveillance (watchlist) and transportation application (person and car detection), respectively. This improvement is achieved while honoring the application performance requirements, and it is over an alternative scheme that employs a static and overprovisioned resource allocation strategy by ignoring the resource coupling relationships.