We present an overview of our studies on some simple mechanical systems including the ‘simple’ nonlinear pendulum and its variants. We show that these systems exhibit numerous types of regular bursting oscillations which are seen in biological neurons. In particular, we discuss bow-tie shaped bursts which we foundin a driven pendulum with linear velocity damping, under constant torque and dynamic feedback. Similar bursts of identical bow-tie shape have been reported by us previously in a system of two resistively coupled Josephson junctions in a certain parameter regime under certain conditions. We discuss the bifurcation mechanism producing some of these bursts.