Consider the parabolic Cauchy problem (1) \(u'(t)-A(t)u(t)=f(t)\), \(t\in[s,T]\), \(u(s)=x\), where the operators \(A(t)\) generate analytic semigroups in a Banach space \(E\) and have possibly non-dense domains. Under some assumptions it can be shown that there exists the evolution operator \(U(t,s)\) for the problem (1). The main object of the paper is a systematic study of the properties of the operator-valued function \(s\to U(t,s)\). Most of the results are obtained under the following hypotheses: (i) for each \(t\in[0,T]\), \(A(t):D_{A(t)}\subset E\to E\) is a closed linear operator with possibly non-dense domain, (ii) there exist \(\alpha_ 0\in(\pi/2,\pi)\) and \(M>0\) such that \(\rho(A(t))\supset S(\alpha_ 0)=\{z\in C:|\arg z|\leq\alpha_ 0\}\cup\{0\}\) and \(\|(\lambda I-A(t))^{-1}\|\leq M/(1+|\lambda|)\) for every \(t\in [0,T]\) and \(\lambda\in S(\alpha_ 0)\), (iii) there exists \(\eta\in(0,1)\) and \(q>0\) such that \(\left\|{d\over dt}A(t)^{-1}- {d\over ds}A(s)^{-1}\right\|\leq q| t-s|^ \eta\) for all \(t,s\in[0,T]\).