A first-order sentence in the language of group theory is a mathematical statement whose variables refer only to elements of a group, and two groups are said to be elementarily equivalent if they satisfy the same first-order sentences. I will introduce these concepts using simple examples, then turn to a question posed by Tarski in the 1940s, known as the Tarski problem: are non-abelian free groups elementarily equivalent? Despite the apparent simplicity of its formulation, this problem remained open until the early 2000s, when it was solved by Sela and by Kharlampovich and Myasnikov.