Active systems refer to a diverse class of non-equilibrium particles that extract energy from the environment, such as chemical energy, to produce mechanical work and directed mot ion. The physics of these systems is, in general, complex and non-intuitive, with rich dynamics and emergent behaviors. We mainly focus on understanding the principles governing active motion in single-particle systems, such as motor proteins and self-propelling particles, using analytical and numerical methods. In the first part of our work, we present a mathematical and computational analysis to estimate protein friction, which constrains the speed and efficiency of motor proteins. We then used a simpler version of this model to analyse motor activity in the context of cargo transport in living cells by computing various dynamical quantities associated with such motion, such as auto-correlation functions for position and orientation of the cargo particles, as well as the cross-correlation between them. Active Brownian particle is a paradigmatic model for active transport, and represents the dynamics of diverse systems such as active colloids and run-and-tumble bacteria. In the third part of our work, we explored the nature of nonequilibrium stationary states of an active Brownian particle in a harmonic potential, and the transitions between these states.