Self-propelled colloids, active polymers and membranes, driven (vibrated) granular layers and hybrid synthetic-biological systems are striking examples of systems containing synthetic active Brownian particles. Such particles autonomously convert the available energy of the environment into their own directed mechanical motion. In most studies the self-propelled Brownian particles move in overdamped media. Recently, experiments with Janus particles in a low-pressure plasma have appeared. A distinctive feature of such a medium is an extremely low viscosity at which the inertial effects play a significant role, resulting in underdamped Brownian motion. At present, there is a lack of statistical theory describing the underdamped Brownian motion of self-propelled particles at all time scales. This paper presents the numerical simulation results of active Brownian motion in homogeneous media of different viscosities. The calculations are performed using a mathematical model of a self-propelled Brownian sphere with translational and rotational inertia. The time-dependent mean square displacement and mean linear displacement (the noise-averaged trajectory) of the particle are investigated as a function of medium viscosity, self-propulsion velocity and moment of inertia. Our simulation reveals that the dynamics of a self-propelled spherical particle significantly depends on two independent dimensionless parameters of the particle: the ratio of the self-propulsion velocity to the characteristic thermal velocity and the ratio of the friction coefficient to the rotational diffusion coefficient. The obtained statistical characteristics of active Brownian motion are compared with the known theoretical models in a wide range of medium viscosities. We propose simple corrections to the basic theory of overdamped active Brownian motion, which allow one to calculate the effective diffusion coefficient and the persistence length of a self-propelled Brownian particle in a medium with any dynamic viscosity. The results obtained are discussed in relation to active particles in a colloidal plasma and superfluid helium.