TITLE: Examples of improved notation that impacted research?
QUESTION [26 upvotes]: The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post Suggestions for good notation. The difference is that I would be interested especially in the practical impact of the improved notation, i.e. examples that have actually created a better understanding of a given topic, or have advanced actual research work on a given topic, or communication about results.
I would be interested in three aspects in particular
(1) Clarity and Insights: Improved and simplified notation that made structures and properties more clearly visible, and enabled insights for the researcher.
(2) Efficiency and Focus: Notation that created efficiencies (e.g., using less space and needed less time, dropped unnecessary or redundant details).
(3) Communication and Exposition: Improved notation that supported communicating and sharing new definitions and results. And notation that evolved and improved in the process of communication. Would you have any practical examples of this evolving process, including dead-ends and breakthroughs?
Edit: Have received great examples in the answers that illustrate what I am interested in. Very grateful for that!

REPLY [3 votes]: I'm a bit surprised that no one mentioned the Einstein summation convention