This little book is the third revised edition of a textbook targeted at university students who have passed calculus courses and are entering the world of pure mathematics. The core of the text is a detailed explanation of basic mathematical concepts and notions, such as sets (up to concepts of denumerability and uncountability), operations on them, functions and relations. Also, mathematical notation is analyzed. There are parts written as an essay (What is mathematics and what does it do for us?) and short historical remarks - these should give a motivation rather than a complete account of the topic. The text is accompanied by a lot of exercises. The style is very narrative, sometimes too much, with the aim to motivate and persuade at every step. Therefore the book should even find interested high-school students among its readers.