This is the third collection under the rubric ''That's Maths''. The first volume, ''That's Maths: The Mathematical Magic in Everyday Life'', was published by Gill Books in 2016. The second, ''That's Maths II: A Ton of Wonders'', was published for Logic Press in 2020. Both volumes were very favourably reviewed. This collection includes essays on a wide range of topics in both pure and applied mathematics. Chapters such as one on the beautiful theory of complex variables, illustrate the ''elegant abstraction'' mentioned in the subtitle, while chapters on mathematics in mechanics, astronomy, networks, freak waves and music demonstrate its ''eclectic applications''. The book is suitable for everyone with an interest in mathematics, whether or not they have studied the subject beyond secondary school level. Some chapters are purely expository; others go deeper into their subjects and are at a more advanced mathematical level. The first three quarters of the book (48 chapters) require no advanced knowledge. The final quarter (16 chapters, each marked by a star) involve some undergraduate maths. These should give readers a flavour of some more advanced mathematics. The chapters in this book are all short, generally about four pages. They can be read in sequence, with constantly changing themes, or in random order. The book should be appreciated by anyone with an interest in mathematics and science. In particular, teachers of these subjects, at both school and university level, should find a cornucopia of material to add a sparkle to their classes. Many of the essays in this collection have appeared in the regular column ''That's Maths'' in The Irish Times or on Peter Lynch's mathematical blog, thatsmaths.com. All have been fully revised for this book.