Network Function Virtualization (NFV) has been emerging as an appealing solution that transforms complex network functions from dedicated hardware implementations to software instances running in a virtualized environment. Due to the numerous advantages such as flexibility, efficiency, scalability, short deployment cycles, and service upgrade, NFV has been widely recognized as the next-generation network service provisioning paradigm. In NFV, the requested service is implemented by a sequence of Virtual Network Functions (VNF) that can run on generic servers by leveraging the virtualization technology. These VNFs are pitched with a predefined order through which data flows traverse, and it is also known as the Service Function Chaining (SFC). In this article, we provide an overview of recent advances of resource allocation in NFV. We generalize and analyze four representative resource allocation problems, namely, (1) the VNF Placement and Traffic Routing problem, (2) VNF Placement problem, (3) Traffic Routing problem in NFV, and (4) the VNF Redeployment and Consolidation problem. After that, we study the delay calculation models and VNF protection (availability) models in NFV resource allocation, which are two important Quality of Service (QoS) parameters. Subsequently, we classify and summarize the representative work for solving the generalized problems by considering various QoS parameters (e.g., cost, delay, reliability, and energy) and different scenarios (e.g., edge cloud, online provisioning, and distributed provisioning). Finally, we conclude our article with a short discussion on the state-of-the-art and emerging topics in the related fields, and highlight areas where we expect high potential for future research.