The architectural framework of Fog and edge computing reveals that the network components which lie between the cloud and devices computes application oriented operations. In this paper, an in-depth review of fog and mist computing in the area of health care informatics is analyzed, classified, and discussed various applications cited in the literature. For that purpose, applications are classified into different categories and a list of application-oriented tasks that can be handled by fog and edge computing are enlisted. It is further added that on which layer of the network system such fog and edge computing tasks can be computed and trade-offs with respect to requirements relevant to healthcare are provided. The review undertaken in this paper focuses on three important areas: firstly, the enormous amount of computing tasks of healthcare system can take mileage of these two computing principles; secondly, the limitation of wireless devices can be overcome by having higher network tiers which can execute tasks and aggregate the data; and thirdly, privacy concerns and dependability prevent computation tasks to completely move to the cloud. Another area which has been considered in the study is how Edge and Fog computing can make the security algorithms more efficient. The findings of the study provide evidence of the need for a logical and consistent approach towards fog and mist computing in healthcare system.