Multibody dynamics and its various applications in vehicle analysis, aerospace engineering, robotics, and biomechanics have experienced a significant development over the last decades. From the numerical analysis point of view, the question of constraints and their discretization is one of the key issues in this field. The first part of the talk concentrates on systems of rigid bodies and the equations of constrained mechanical motion. The properties of these differential-algebraic equations (DAEs) are discussed and an algorithm for real-time applications with hardware-in-the-loop features is introduced. A simulation of a vehicle with trailer illustrates the approach. In the second part of the talk, deformable bodies are included in terms of a dynamic saddle point problem as basic mathematical model. It turns out that the DAE index and the inf-sup condition are closely related here. Moreover, a combination of the HHT and the RATTLE time integrators is applied to the method of lines and its reversed counterpart. The talk closes with simulations of pantograph and catenary dynamics.