Although academic and industry experts are now advocating for going from large-centralized Cloud Computing infrastructures to smaller ones massively distributed at the edge of the network, management systems to operate and use such infrastructures are still missing. In this paper, we focus on the monitoring service which is a key element to any management system in charge of operating a distributed infrastructure. Several solutions have been proposed in the past for cluster, grid and cloud systems. However, none is well appropriate to the Fog/Edge context. Our goal in this study, is to pave the way towards a holistic monitoring service for a Fog/Edge infrastructure hosting next generation digital services. The contributions of our work are: (i) the problem statement, (ii) a classification and a qualitative analysis of major existing solutions, and (iii) a preliminary discussion on the impact of the deployment strategy of functions composing the monitoring service.