Multi-access Edge Computing (MEC) is a new paradigm that brings storage and computing close to the clients. MEC enables the deployment of complex network-assisted mechanisms for video streaming that improve clients' Quality of Experience (QoE). One of these mechanisms is segment prefetching, which transmits the future video segments in advance closer to the client to serve content with lower latency. In this work, for HAS-based (HTTP Adaptive Streaming) video streaming and specifically considering a cellular (e.g., 5G) network edge, we present our approach Segment Prefetching and Caching at the Edge for Adaptive Video Streaming (SPACE). We propose and analyze different segment prefetching policies that differ in resource utilization, player and radio metrics needed, and deployment complexity. This variety of policies can dynamically adapt to the network's current conditions and the service provider's needs. We present segment prefetching policies based on diverse approaches and techniques: past segment requests, segment transrating (i.e., reducing segment bitrate/quality), Markov prediction model, machine learning to predict future segment requests, and super-resolution. We study their performance and feasibility using metrics such as QoE characteristics, computing times, prefetching hits, and link bitrate consumption. We analyze and discuss which segment prefetching policy is better under which circumstances, as well as the influence of the client-side Adaptive Bit Rate (ABR) algorithm and the set of available representations ("bitrate ladder") in segment prefetching. Moreover, we examine the impact on segment prefetching of different caching policies for (pre-)fetched segments, including Least Recently Used (LRU), Least Frequently Used (LFU), and our proposed popularity-based caching policy Least Popular Used (LPU).INDEX TERMS Adaptive video streaming, content delivery, HAS, edge computing, cellular network edge, MEC, segment prefetching, segment caching.