Linear Volterra-Stieltjes differential equations are considered. We give a sufficient condition which ensures that the uniform asymptotic stability and the exponential asymptotic stability of linear Volterra-Stieltjes differential equations coincide. In particular, this yields necessary and sufficient conditions for the exponential asymptotic stability of linear Volterra-Stieltjes differential equations whose kernels are monotone. Then the class of positive linear Volterra-Stieltjes differential equations is studied in detail and simple criteria for the exponential asymptotic stability of positive equations are presented. Furthermore, some sufficient conditions for the exponential asymptotic stability of positive equations subjected to structured perturbations and affine perturbations are provided. Finally, by applying obtained results to linear Volterra integro-differential equations with infinite delay, we get explicit criteria for the exponential asymptotic stability of positive equations with delay. To the best of our knowledge, most of the obtained results of this paper are new.