This unique book gives an informal introduction into the non-Euclidean geometries through a series of dialogues between a somewhat grown-up Alice (of Looking Glass fame), her uncle Lewis Carroll, and a visitor from the twentieth century, Dr Whatif. In the story, Lewis Carroll's geometrical beliefs are cast into the Euclidean mould, Dr. Whatif asks the penetrating and controversial questions, and Alice acts as a mediator and interested participant. The book is intentionally more mathematical than Lewis Carroll's books, but for those of us who enjoyed Alice's earlier adventures, there are many interesting flashbacks to those inimitable characters: the Red Queen, Tweedle-Dum and his twin brother, the Mad Hatter...The text is filled with humour, wit, and verses of poetry. Part 1 contains the story in six chapters, each of which concludes with a problem set; Part 2 is more mathematical, and looks at the axiom systems, and gives solutions to the problems. The presentation, with its old-time borders, script headings, and cartoon drawings evokes the spirit of the original Alice.