In a sequence of papers the author has obtained important results on codimensions and cocharacters of T-ideals over a field of characteristic zero. The main theorem in the paper under review claims that the codimensions of the \(k\times k\) matrix algebra \(F_ k\) are asymptotically equal to the trace codimensions: \(c_ n(F_ k)\cong t_ n(F_ k)\cong s(n)k^{2n},n\to \infty\), where s(n) is a known rational function. As a corollary, the asymptotic behaviour of the codimensions is established for the Capelli identity in \(k^ 2+1\) skew symmetric variables.