In modern 5G networks and beyond, finite blocklength lossy source coding is essential to provide ultra-reliable and low-latency communications. The analysis of point-to-point and multiterminal settings from the perspective of finite blocklength lossy source coding is therefore of great interest to 5G system designers and is also related to other long-standing problems in information theory. In this monograph, the authors provide a comprehensive review of recent advances in second-order asymptotics for lossy source coding. The monograph is divided into three parts. In Part I, the reader is introduced to the topic with a description of the mathematical tools used and a basic example of lossless source coding. In Part II, the authors present five generations of the rate-distortion problem to tackle various aspects of practical quantization tasks: noisy source, noisy channel, mismatched code, Gauss-Markov source and fixed-to-variable length compression. In Part III, the authors present four multi-terminal generalizations of the rate-distortion problem to consider multiple encoders, decoders or source sequences: the Kaspi problem, the successive refinement problem, the Fu-Yeung problem and the Gray-Wyner problem. This self-contained, complete review of a topic at the heart of 5G technology is essential reading for all students, researchers and designers of modern day communication systems.