This self-contained graduate text provides a readily available source of information and reference for a substantial part of ring theory developed over the last 80 years. The book completes a trilogy with the author’s A First Course in Noncommutative Rings and Lectures on Modules and Rings. It deals with a number of current topics in noncommutative algebra, ranging from von Neumann regular rings, unit-regular rings and strongly regular rings, exchange (or "suitable") rings, to clean, strongly clean, and nil-clean rings, with a dosage of the popular theory of generalized inverses (including Drazin inverses and Moore–Penrose inverses) in between. Following all of the above ring-theoretic "excursions", the text concludes with a final chapter on the issues of cancellation and substitution of modules, along with a discussion on the relatively classical stable range  theory of rings. The rich exercise sets for the various chapters of the book are designed to be especially helpful for students and researchers alike who desire to deepen their understanding of noncommutative algebra.