We present the definitions of two new Lyapunov functionals. These functionals are applied to establish sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integrodifferential equations of the first order. The obtained results improve and extend known results available from literature. We also propose examples to show the applicability of our results and illustrate them. By using MATLAB-Simulink, we clearly show, in particular cases, the behavior of orbits of the analyzed Volterra integrodifferential equations.