The second volume of the book [for a review of the first one see Zbl 0751.00002)] of prominent Russian algebraists contains four chapters: semigroups, lattices, universal algebras, categories. This book should be treated as a guide for non-specialists on the mentioned branches of algebra. Definitions, notions, fundamental theorems together with new results, from the last twenty years (approximately) are presented. Each chapter ends with references of textbooks and about 50 current articles. The material of the book consists of things which can be applied in other areas of mathematics, e.g., linguistics, codes, information.