For a finite simple graph \(G\), a strong edge colouring of \(G\) is an edge colouring in which every colour class is an induced matching. (Since each class is a matching, the colouring is proper.) The strong chromatic index of \(G\), \(\chi_{s}(G)\), is the smallest number of colours in a strong edge colouring of \(G\). The paper under review studies the strong chromatic index of random graphs \(G(n,p(n))\) where each edge arises with probability \(p(n)\) independently of all other edges. \textit{Z. Palka} [Australas. J. Comb. 18, 219-226 (1998; Zbl 0923.05042)] showed that if \(p(n)=\Theta(n^{-1})\), then \[ \lim_{n\rightarrow\infty}P\{\chi_{s}(G(n,p(n)))=O(\Delta(G(n,p(n))))\}=1 \] where, as usual, \(\Delta=\Delta(G)\) is the maximum degree of the graph. \textit{V. H. Vu} [Comb. Probab. Comput. 11, 103--111 (2002; Zbl 0991.05041)] used Rödl-nibble type arguments to show that if \(\log(n)^{1+\delta}/n\leq p(n)\leq n^{-\varepsilon}\) for any \(\varepsilon>0\) and \(\deltan^{-\varepsilon}\) for all \(\varepsilon>0\). The main result is that \[ \lim_{n\rightarrow\infty}P\{\frac{(1-o(1))p{n\choose 2}}{\log_{1/(1-p)}(n)}\leq \chi_{s}(G(n,p(n)))\leq \frac{(2+o(1))p{n\choose 2}}{\log_{1/(1-p)}(n)}\}=1. \] The lower bound is easy: a typical \(G(n,p)\) has about \(p{n\choose 2}\) edges and it is easy to see, using the first moment method, that the largest induced matching in \(G(n,p)\) has about \(\log_{1/(1-p)}(n)\) edges. More interesting part is the upper bound, derived from the result of \textit{B. Bollobás} [Combinatorica 8, 49--56 (1988; Zbl 0666.05033)] to the effect that the (ordinary) vertex chromatic number \(\chi(G(n,p))\) for \(0