With the growing demand for openness, scalability, and granularity, mobile network function virtualization (NFV) has emerged as a key enabler for the most of mobile network operators. NFV decouples network functions from hardware devices. This decoupling allows network services, called Virtualized Network Functions (VNFs), to be hosted on commodity hardware which simplifies and enhances service deployment and management for providers, improves flexibility, and leads to efficient and scalable resource usage, and lower costs. The proper placement of VNFs in the hosting infrastructures is one of the main technical challenges. This placement significantly influences the network’s performance, reliability, and operating costs. The VNF placement is NP-Hard. Therefore, there is a need for placement methods that can cope with the complexity of the problem and find appropriate solutions in a reasonable duration. The primary purpose of this study is to provide a taxonomy of optimization techniques used to tackle the VNF placement problems. We classify the studied papers based on performance metrics, methods, algorithms, and environment. Virtualization is not limited to simply replacing physical machines with virtual machines or VNFs, but may also include micro-services, containers, and cloud-native systems. In this context, the second part of our article focuses on the placement of Containers Network Functions (CNFs) in edge/fog computing. Many issues have been considered as traffic congestion, resource utilization, energy consumption, performance degradation, etc. For each matter, various solutions are proposed through different surveys and research papers in which each one addresses the placement problem in a specific manner by suggesting single objective or multi-objective methods based on different types of algorithms such as heuristic, meta-heuristic, and machine learning algorithms.