A general wave approach for the vibration analysis of curved beam structures is presented. The analysis is based on wave propagation, transmission, and reflection, including the effects of both propagating and decaying near-field wave components. A matrix formulation is used that offers a systematic and concise method for tackling free and forced vibrations of complex curved beam structures. To illustrate the effectiveness of the approach, several numerical examples are presented. The predictions made using the wave approach are shown to be in excellent agreement with a conventional finite element analysis, with the advantage of reduced computational costs and good conditioning number of the characteristic equation. The developed wave approach is applied to investigate the free vibration, vibration transmission, and power flow of built-up structures consisting of curved beams, straight beams, and masses, with the aim for designing vibration isolation structure with high attenuation ability. Wave reflection and transmission in the infinite curved beam structure, as well as vibration and energy transmission in coupled finite curved beam structure are investigated. Numerical results show that wave mode conversion takes place for the reflected and transmitted wave propagating through a curved beam, and the power flow in the coupled curved beam structure shows energy attenuation and conversion by curved beam and the discontinuities. The investigation will shed some light on the designing of curved beam structures.