By using Sadovskij's fixed point theorem, the author studies the controllability and the local controllability of the following non-autonomous neutral functional differential systems with unbounded delay \[ {d\over dt}[x(t)+F(t,x_{t})]+A(t)x(t)=Cu(t)+G(t,x_{t}), \quad 0\leq t\leq a \] \[ x_{0}=\phi\in {\mathcal B} \] where the state variable \(x(\cdot)\) takes values in Banach space \(X\) and the control function \(u(\cdot)\) is given in \(L^ 2([0,a],U),\) a Banach space of admissible control functions, with \(U\) as a Banach space. \(C\) is a bounded linear operator from \(U\) into \(X,\) the family \(\{A(t): 0\leq t\leq a\}\) of unbounded linear operators generates a linear evolution system, \(F,G: [0,a]\to {\mathcal B}\) are appropriate functions, and \({\mathcal B}\) is the phase space defined axiomatically and introduced by \textit{J. K. Hale} and \textit{J. Kato} [Funkc. Ekvacioj, Ser. Int. 21, 11--41 (1978; Zbl 0383.34055)]. As application, an example is provided to illustrate the obtained results. The results of this paper extend previous results of the author [Appl. Math. Comput. 141, 281--296 (2003; Zbl 1175.93029)] concerning the autonomous case \(A(t)\equiv A.\)