This document surveys recent and current developments in Low Earth Orbit (LEO) satellite networking. It presents a brief overview of satellite networking in order to contextualize the issue. It then focuses on current research work in emerging domains, such as machine learning, Software-Defined Networking (SDN), low latency networking, green networking, Information-Centric Networks (ICN), etc. For each of these, it presents recent works and a direction of the research community within that emerging domain. The paper also describes the current state of standardization efforts in 3GPP and in IETF for LEO satellite networking. In particular, we present in some detail the direction these standards bodies are pointing towards for LEO networking with inter-satellites links. Finally, some future challenges and interesting research directions are described and encouraged. This is an overview of the current state of the LEO satellite research in both academic and industrial standardization environments which we believe will be helpful to understand the current state of the art.