The author gives an excellent account of some of the main ideas and techniques for treating, from an abstract point of view, systems of linear equations of the form \[ \begin{matrix}\dot u_1(t) & = & A_{11}u_1(t) +\cdots+ A_{1n}u_n(t),\;u_1(0) & = & x_1\\ &\vdots &&\vdots\\ \dot u_n(t) &= & A_{n1}u_1(t) +\cdots+ A_{nn}u_n(t),\;u_n(0) &=& x_n.\end{matrix}\tag{1} \] Here each \(A_{ij}\) is a linear, in general unbounded operator between Banach spaces \(E_j\) and \(E_i\). Expressed in terms of the operator matrix \({\mathcal A}=(A_{ij})_{n\times n}\) defined on the product space \({\mathcal E}= E_1\times E_2\times\cdots\times E_n\), (1) becomes the abstract Cauchy problem \[ \dot{\mathbf u}(t)= {\mathcal A}{\mathbf u}(t),\quad t\geq 0,\quad{\mathbf u}(0)={\mathbf x},\tag{2} \] and the aim is then to establish the well-posedness of (2) by applying the theory of strongly continuous semigroups. The author highlights the extreme difficulties involved in finding conditions on the entries \(A_{ij}\) for \(\mathcal A\) to be the generator of a strongly continuous semigroup on \(\mathcal E\). However, he then goes on to explain how these difficulties can be overcome for certain restricted classes of operator matrices that appear frequently in applications. In particular, the cases of polynomial operator matrices, matrix multiplicators, operator matrices with diagonal domains, and systems arising from complete second-order Cauchy problems are all discussed. A comprehensive list of more than seventy references is provided.