Let \({\mathcal A}\), \({\mathcal B}\) be real or complex algebras. A sequence \(H=\{h_0,h_1,\dots\}\) of additive operators from \({\mathcal A}\) to \({\mathcal B}\) is called a \textit{higher ring derivation} if \[ h_n(zw)=\sum_{i=0}^{n}h_i(z)h_{n-i}(w),\qquad z,w\in{\mathcal A}, n=0,1,\dots. \] A sequence \(F=\{f_0,f_1,\dots\}\) of operators from \({\mathcal A}\) to \({\mathcal B}\) is called a \textit{higher derivation} if \[ f_n(x+y+zw)=f_n(x)+f_n(y)+\sum_{i=0}^{n}f_i(z)f_{n-i}(w),\qquad x,y,z,w\in{\mathcal A}, n=0,1,\dots. \] The main goal of the paper is to consider approximate higher derivations and the problem of the stability of higher ring derivations. It is shown, in particular, that if a sequence \(F=\{f_0,f_1,\dots\}\) satisfies, with some given control mappings \(\varphi_n:{\mathcal A}^4\to[0,\infty)\), \[ \|f_n(x+y+zw)-f_n(x)-f_n(y)-\sum_{i=0}^{n}f_i(z)f_{n-i}(w)\|\leq\varphi_n(x,y,z,w) \] for all \(x,y,z,w\in {\mathcal A}\) and \(n=0,1,\dots\), then there exists a unique higher ring derivation \(H=\{h_0,h_1,\dots\}\) such that \(h_n\) is somehow \textit{close} to \(f_n\) for each \(n\). Several corollaries are obtained for particular control mappings \(\varphi_n\) and under some additional assumptions upon \({\mathcal A}\). The results refer in particular to \textit{D. G. Bourgin} [Duke Math. J. 16, 385--397 (1949; Zbl 0033.37702)], \textit{R. Badora} [Math. Inequal. Appl. 9, No.~1, 167--173 (2006; Zbl 1093.39024)], \textit{T. Miura}, \textit{G. Hirasawa} and \textit{S.-E. Takahasi} [J. Math. Anal. Appl. 319, No.~2, 552--530 (2006; Zbl 1104.39025)].