Future wireless networks will provide high bandwidth, low-latency, and ultra-reliable Internet connectivity to meet the requirements of different applications, ranging from mobile broadband to the Internet of Things. To this aim, mobile edge caching, computing, and communication (edge-C3) have emerged to bring network resources (i.e., bandwidth, storage, and computing) closer to end users. Edge-C3 allows improving the network resource utilization as well as the quality of experience (QoE) of end users. Recently, several video-oriented mobile applications (e.g., live content sharing, gaming, and augmented reality) have leveraged edge-C3 in diverse scenarios involving video streaming in both the downlink and the uplink. Hence, a large number of recent works have studied the implications of video analysis and streaming through edge-C3. This article presents an in-depth survey on video edge-C3 challenges and state-of-the-art solutions in next-generation wireless and mobile networks. Specifically, it includes: a tutorial on video streaming in mobile networks (e.g., video encoding and adaptive bitrate streaming); an overview of mobile network architectures, enabling technologies, and applications for video edge-C3; video edge computing and analytics in uplink scenarios (e.g., architectures, analytics, and applications); and video edge caching, computing and communication methods in downlink scenarios (e.g., collaborative, popularity-based, and context-aware). A new taxonomy for video edge-C3 is proposed and the major contributions of recent studies are first highlighted and then systematically compared. Finally, several open problems and key challenges for future research are outlined.