Fog computing is a new computational paradigm that emerged from the need to reduce network usage and latency in the Internet of Things (IoT). Fog can be considered as a continuum between the cloud layer and IoT users that allows the execution of applications or storage/processing of data in network infrastructure devices. The heterogeneity and wider distribution of fog devices are the key differences between cloud and fog infrastructure. Genetic-based optimization is commonly used in distributed systems; however, the differentiating features of fog computing require new designs, studies, and experimentation. The growing research in the field of genetic-based fog resource optimization and the lack of previous analysis in this field have encouraged us to present a comprehensive, exhaustive, and systematic review of the most recent research works. Resource optimization techniques in fog were examined and analyzed, with special emphasis on genetic-based solutions and their characteristics and design alternatives. We defined a classification of the optimization scope in fog infrastructures and used this optimization taxonomy to classify the 70 papers in this survey. Subsequently, the papers were assessed in terms of genetic optimization design. Finally, the benefits and limitations of each surveyed work are outlined in this paper. Based on these previous analyses of the relevant literature, future research directions were identified. We concluded that more research efforts are needed to address the current challenges in data management, workflow scheduling, and service placement. Additionally, there is still room for improved designs and deployments of parallel and hybrid genetic algorithms that leverage, and adapt to, the heterogeneity and distributed features of fog domains.