Novel computing paradigms aim to enable better hardware utilization, allowing a greater number of applications to be executed on the same physical resources. Serverless computing is one example of such an emerging paradigm, enabling faster development, more efficient resource usage, as well as no requirements for infrastructure management by end users. Recently, efforts have been made to utilize serverless computing at the network edge, primarily focusing on supporting Internet of Things (IoT) workloads. This study explores open issues, outlines current progress, and summarizes existing research findings about serverless edge computing for IoT by analyzing 67 relevant papers published between 01.01.2015 and 01.09.2021. We discuss the state-of-the-art research in 8 subject areas relevant to the use of serverless at the network edge, derived through the analysis of the selected articles. Results show that even though there is a noticeable interest for this topic, further work is needed to adapt serverless to the resource constrained environment of the edge.