The authors apply the Kurzweil-Henstock integral formalism to give existence theorems for linear Volterra equations \[ x(t)+^{\ast}\int_{[a,t]}\alpha(s)x(s)\,ds=f(t),\qquad t\in[ a,b],\tag{1} \] where the functions \(x,f\)\ have values in the Banach space \(X\). For example, existence of the solution is proved for the above equation with Henstock integral when \(\alpha\in H\left( [a,b],L(X)\right) \)\ is bounded, and \(x,f\in BV_{a}^{-}\left( [a,b],X\right) \)\ (respectively \(\alpha\)\ is absolutely integrable and \(x,f\in C\left( [a,b],X\right) \)) under some additional hypotheses.