The paper is a brief survey on the stability of isometries on real Banach spaces. An \(\varepsilon\)-isometry between two Banach spaces \(X,Y\) is a map \( f:X\to Y \) satisfying \( |\|f(x)-f(y)\|- \|x-y\||\leq \varepsilon, \forall x,y\in X.\) For an isometry \(U:X\to Y \) let dist\((f,U)=\inf\{\|f(x)-U(x)\|:x\in X\}.\) The paper is concerned with the problem of the existence of a constant \(K(X,Y)\) such that for every surjective \(\varepsilon\)-isometry \( f:X\to Y \) satisfying \(f(0) =0\) there exists a surjective linear isometry \(U:X\to Y\) with dist\((f,U) \leq K(X,Y)\varepsilon.\) Another problem is to find the best constant \(K(X,Y).\) The author and \textit{M. Omladič} [Math. Ann. 303, 617-628 (1995; Zbl 0836.46014)], proved that \( K(X,Y)=2\) works for all real Banach spaces and that this estimation is sharp. Denoting by \(\mathcal U\) the set of all surjective isometries of \(X\) onto \(Y\) (by Mazur-Ulam theorem they must be affine) then \(\text{dist}(f,\mathcal U):= \inf\{\text{dist}(f,U)\); \(U\in \mathcal U\} \leq \gamma_X \varepsilon,\) where \(\gamma_X\) is the Jung constant of the space \(X\) [see \textit{D. Amir}, Pac. J. Math. 118, 1-15 (1985; Zbl 0529.46011)]. This is a result of the author to appear in Houston J. Math. The surjectivity hypothesis in the above result can be weakened provided the points of Fréchet differentiability of the norm of \(X\) is dense. Two open problems are posed at the end of the paper.