  This article presents an in-depth educational overview of the latest
mathematical developments in coupled cluster (CC) theory, beginning with
Schneider's seminal work from 2009 that introduced the first local analysis of
CC theory. We offer a tutorial review of second quantization and the CC ansatz,
laying the groundwork for understanding the mathematical basis of the theory.
This is followed by a detailed exploration of the most recent mathematical
advancements in CC theory.Our review starts with an in-depth look at the local
analysis pioneered by Schneider which has since been applied to analyze various
CC methods. We then move on to discuss the graph-based framework for CC methods
developed by Csirik and Laestadius. This framework provides a comprehensive
platform for comparing different CC methods, including multireference
approaches. Next, we delve into the latest numerical analysis results analyzing
the single reference CC method developed by Hassan, Maday, and Wang. This very
general approach is based on the invertibility of the CC function's Fr\'echet
derivative. We conclude the article with a discussion on the recent
incorporation of algebraic geometry into CC theory, highlighting how this novel
and fundamentally different mathematical perspective has furthered our
understanding and provides exciting pathways to new computational approaches.
