The subject of this research is the improvement of the dynamic behaviour of structures using passive and active approaches to vibration control by using inerters. Inerter is a relatively new element in the theory of mechanical networks. It is a mechanical device that generates force proportional to the relative acceleration between its terminals. The use of inerters is still relatively unexplored and offers many new possibilities of reducing unwanted vibration effects, which is of particular importance in the resonant working conditions of the structure. In the first part of the thesis, passive linear dynamic systems are considered. In the second part, the passive systems are enhanced by the active control. Stability analysis of the active systems is performed. In addition to the parameters that ensure stable operation of the system, the optimization of vibration behaviour of active structures is carried out. The main optimization criterion used is the minimization of the specific kinetic energy of system vibration in a broad frequency band. Throughout the work, analytical and numerical methods are combined, depending on the complexity of the considered system. The achieved results are compared through discussion on the usefulness of the active control and inerter implementation, depending on the system considered. The motivation for the work are synergistic effects regarding the utilization of inerters in parallel with active control systems. The result of the research is a development of new methods for passive and active vibration control. Furthermore, it is shown that employing the inerter in isolation systems can yield with substantial improvements in fatigue life of isolator coupling components, i.e. springs. Novel cylindrical helical spring stress and displacement correction factors are proposed.