Abstract. In this thesis we study the non-smoothness dynamical systems using a suite of numerical, analytical and experimental methods. First the concept of non-smoothness is introduced by dividing the global space into a finite number of smooth subspaces. If a dynamical system is linear in all these subspaces then an implicit global analytical solution can be given providing the time occurrences of non-smoothness are determined. Mathematically this leads to a set of strongly nonlinear algebraic equations. To illustrate the concept and demonstrate the methodology of solving the non-smooth dynamical systems, four different engineering problems are studied theoretically and on purpose built experimental rigs. Firstly, the impacting and frictional oscillators are investigated and the [mdings are used to study two engineering systems: ground moling and percussive ultrasonic drilling of hard materials. The governing dynamics is analysed and the results are used to maximise the progression rates. The second example comes from rotordynamics, where the nonlinear interactions between a rotor and its snubber ring are investigated. The results obtained from the developed mathematical model confronted with the experiments shown good correlation. The non-smoothness in the third problem relates to the opening and closing of a fatigue crack. In this study a novel fatigue testing rig was built and its mathematical model was developed. The results from the theoretical modelling correlate well with the experimental [mdings. In the last application, the dynamics of metal cutting focusing on the frictional chatter is investigated. It has been shown that the bifurcation analysis can be instrumental in choosing the system parameters to avoid chatter. The list of references includes the major works in the area followed by the publications constituting this thesis.