Inspiring popular video games like Tetris while contributing to the study of combinatorial geometry and tiling theory, polyominoes have continued to spark interest ever since their inventor, Solomon Golomb, introduced them to puzzle enthusiasts several decades ago. In this expanded edition, the author takes a new generation of readers on a mathematical journey into the world of polyominoes, incorporating important recent developments. Deceptively simple, polyominoes are a collection of squares joined together along their edges. But how many different polyominoes can you make with 5 squares, 6 squares, n squares? Posing problems and giving answers along the way, Golomb invites the reader to play with these mathematical structures and develop an understanding of their extraordinary properties. In this new edition, he addresses the properties of octominoes and enneominoes and the problem of how to cover a doughnut with polyominoes. An extensive bibliography has been included to guide the reader to other interesting mathematical conundrums and to more advanced mathematical theories of polyominoes.