Analogy is a quintessential human cognitive capability that supports knowledge transfer across domains. While researchers have long studied analogy generation, evaluation, and reasoning through symbolic and connectionist approaches, the rise of large language models (LLMs) has introduced new generation systems and evaluation methods. However, a systematic understanding of this emerging landscape and its connection to prior work remains underexplored. To address this gap, we conduct a PRISMA-based systematic literature review spanning four decades across six databases. We identify three key observations: first, generation and evaluation methods are strongly shaped by analogy granularity, such as lexical versus compositional analogies; second, human-in-the-loop design plays an important role in LLM-based analogy generation systems; and third, the field remains divided over whether LLMs perform genuine analogical reasoning. Based on our review, we propose practical guidelines for future work on analogy generation, including potential evaluation dimensions for lexical and compositional tasks.