Cables are widely used in cable-stayed bridges and other civil engineering structures, but they often experience large amplitude vibrations due to their low inherent damping. Compared with conventional viscous dampers, inerter-based vibration absorbers can potentially be more effective to improve damping performance. The inerter is a two-port mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals, with the constant of proportionality termed inertance. This thesis is focused on establishing a systematic methodology for identifying optimum inerter-based cable vibration absorbers. In order to facilitate the investigation of a wide range of candidate absorber layouts, a finite element (FE) taut cable model with a generic vibration absorber represented by its admittance function, is firstly established. Then, three performance measures and an optimisation approach used in this study are introduced. Based on the established model, the effects of different absorber layouts for cable vibration suppression are investigated. First, potential advantages of all layouts with no more than one inerter, one damper and one spring each, are investigated. Compared with viscous damper only, the threeelement inerter-based layouts can provide significant performance improvements with large inertance. However, large inertance leads to difficulties in terms of physical implementation. In order to limit the inertance, while maintaining the performance gain, alternative inerter-based configurations containing more elements are studied. Based on network synthesis theory, a pair of fixed-sized-inerter (FSI) layouts are introduced which cover a set of seven-element network layouts with one inerter and at most six other elements (springs and dampers). The results show that one type of FSI layout can provide much more beneficial results than all those low-complexity layouts (with three elements or fewer), with much smaller inertance. Moreover, a simplification procedure is adopted with which two four-element inerter-based layouts are obtained, with no comprise of the performance for specific ranges of inertance values. For the identified beneficial layouts, the effects of series compliance at connections are examined, due to the fact that the connections at either end of the absorber are not fully rigid. Results show that without re-optimisation, the series compliance is detrimental, if properly retuned for specific inertance, even better performance can be obtained in some cases. The effects of the installation location of the absorber on damping performance are also investigated for same identified beneficial layouts. Significant influence of location parameter on damping performance has been found. The study results can be useful for inerter-based absorber for cable vibration problems in engineering application, e.g., in design, tuning and installation of absorbers. Besides, the proposed methodology in this study can be applied to cable vibration problems with other performance criteria, and also other mechanical structures.