In the paper of \textit{D. Popa} [J. Math. Anal. Appl. 309, No. 2, 591--597 (2005; Zbl 1079.39027)] the Hyers-Ulam stability problem was proved for linear recurrences in a Banach space. In the paper under review, the authors investigate this problem for nonlinear recurrences in a metric space \((X, d)\). More precisely, they show that if \(\{x_n\}\), \(\{a_n\}\) and \(\{\varepsilon_n\}\) are sequences in \(X\) , \(X^X\) and \({\mathbb R}_+\), respectively, \(d(x_{n+1}, a_n(x_n))\leq \varepsilon_n \quad (n\in {\mathbb N}_0:={\mathbb N}\cup \{0\})\), there exists \(\{\lambda_n\}\subseteq[0,\infty)\) with \(\limsup_{n\rightarrow \infty}\frac{\varepsilon_{n-1}\lambda_n}{\varepsilon_n}<1\) and \(d(a_n(x),a_n(y))\leq \lambda_nd(x, y) \quad (x, y \in X, n\in{\mathbb N}_0)\), then there exists a sequence \(\{y_n\}\subseteq X\) and a positive constant \(M\) such that \(y_{n+1}=a_n(y_n) \quad (n\in {\mathbb N}_0)\) and \(d(x_n, y_n)\leq M\varepsilon_{n-1} \quad (n\in {\mathbb N})\).