The present paper is devoted to stability properties of the zero solution to the following linear Volterra integro-differential equation \[ x'= A(t)x+ \int_0^t C(t,s) x(s)ds, \quad t\geq 0,\tag{1} \] where \(x(t)\in C^1 (\mathbb{R}_+,\mathbb{R}^n)\), \(A(t)\in C(\mathbb{R}_+,\mathbb{R}^{n\times n})\), \(C(t,s)\in C(\mathbb{R}_+^2, \mathbb{R}^{n\times n})\) for \(0\leq s\leq t<\infty\).
Applying an equivalent form of (1) given by K. N. Murty, M. A. S. Srinivas and V. A. Narasimham [Tamkang J. Math. 19, No. 1, 29-36 (1988; Zbl 0684.45006)], the authors construct a Lyapunov-type functional and use it to obtain some sufficient conditions which ensure stability, uniform stability, asymptotic stability and instability of the zero solution of (1). Some known results of Murty, Srinivas and Narasimham [loc. cit.] and of M. Wang, L. Wang and X. Du [Acta Math. Appl. Sin. 15, No. 2, 184-193 (1992; Zbl 0762.45006)] and of L. Wang and X. Du [Acta Math. Appl. Sin. 15, No. 2, 260-268 (1992; Zbl 0763.45005)] are generalized and improved.