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The ability to offer low latency communications is one of the critical design requirements for the upcoming 5G era. The current practice for achieving low latency is to overprovision network resources (e.g., bandwidth and computing resources). However, this approach is not cost-efficient, and cannot be applied in large-scale. To solve this, more cost-efficient resource management is required to dynamically and efficiently exploit network resources to guarantee low latencies. The advent of network virtualization provides novel opportunities in achieving cost-efficient low latency communications. It decouples network resources from physical machines through virtualization, and groups resources in the form of virtual machines (VMs). By doing so, network resources can be flexibly increased at any network locations through VM auto-scaling to alleviate network delays due to lack of resources. At the same time, the operational cost can be largely reduced by shutting down low-utilized VMs (e.g., energy saving). Also, network virtualization enables the emerging concept of mobile edge-computing, whereby VMs can be utilized to host low latency applications at the network edge to shorten communication latency. Despite these advantages provided by virtualization, a key challenge is the optimal resource management of different physical and virtual resources for low latency communications. This thesis addresses the challenge by deploying a novel cost-efficient resource management framework that aims to solve the cost-efficient design of 1) low latency communication infrastructures; 2) dynamic resource management for low latency applications; and 3) fault-tolerant resource management. Compared to the current practices, the proposed framework achieves 80% of deployment cost reduction for the design of low latency communication infrastructures; continuously saves up to 33% of operational cost through dynamic resource management while always achieving low latencies; and succeeds in providing fault tolerance to low latency communications with a guaranteed operational cost.