Recently, a lot of papers which are concerned with differential equations with infinite delays have appeared. In the present paper, the author first investigates the asymptotic behaviour of solutions for the following linear system: \[ x'(t)=A(t)x(t)+B(t)x(t-\gamma (t)),\quad t\geq t_ 0\geq 0, \] where x(t) is an n-dimensional column vector; A(t), B(t) are real-valued \(n\times n\) matrices; locally integrable on \([t_ 0,+\infty)\); \(\gamma\) (t) is a real-valued, continuous scalar function on \([t_ 0,+\infty)\); \(r(t)>0\), r(t) may be unbounded, but \(t-r(t)\nearrow +\infty\) as \(t\to +\infty\). (Obviously, in this case \(s=t-r(t)\) has a single-valued inverse function \(\alpha\) (s) for \(s\geq t_ 0-r(t_ 0)\) and \(\alpha (s)>s.)\) Furthermore, the author generalizes the above result to the case of several infinite delays and to nonlinear systems as well.