By elementary algebraic/arithmetic reasoning \textit{L. Kronecker} [Berl. Ber. 1884, 1179--1193 (1884; JFM 16.0083.02)] proved: Theorem A. Let \(A\) be an \(N \times M\) matrix with real entries, and let \(\alpha\in \mathbb R^N\) . Then the following two assertions are equivalent: 1. For every \(\varepsilon > 0\) there exists a point \(x \in \mathbb R^M\) such that each of the \(N\) coordinates of the vector \(Ax-\alpha\) are within \(\varepsilon\) of an integer. 2. If \(u \in \mathbb Z^N\) is a lattice point such that \(uA = 0\), then \(u \cdot\alpha = 0\). The authors review the history of this basic result concerning inhomogeneous Diophantine approximation, they mention the proofs of Hardy and Littlewood, Weyl, Landau, Hardy and Wright, Skolem, Mahler and others. The role of Chebyshev's and Fejér's extremal trigonometric polynomials are described. They discuss in detail \textit{P. Turán}'s localized and quantitative Kronecker theorem [Acta Math. Acad. Sci. Hung. 10, 277--298 (1959; Zbl 0103.04503)] and \textit{Y. Chen}'s quantitative Kronecker theorem [Proc. Am. Math. Soc. 123, No. 11, 3279--3284 (1995; Zbl 0855.11034)].