In the last years, elastic wave propagation has become a relevant topic in the field of nonlocal continuum theories for small-size structures. Many works have focused on wave propagation in beams modelled by several approaches, such as Eringen's differential theory, stress and strain gradient theories, higher order nonlocal strain gradient theory, the modified couple stress theory. Few works, instead, have been dedicated to wave propagation in beams modelled by the stress-driven nonlocal theory, adopting the Euler-Bernoulli beam model [1] or the Rayleigh beam model [2]. Recent studies have addressed not only bare beams but also periodic ones, with periodicity due to alternate materials, attached resonators or equally spaced supports. In this context, this study is devoted to the wave propagation analysis of beams proposing a stress-driven nonlocal Timoshenko beam formulation and developing an analytical/computational framework for bare and periodic beams [3]. Specifically, the dispersion curves of the bare beam are obtained analytically for the infinite beam providing a clear perspective on the effects of nonlocality. Wave propagation in periodic beams, either equipped with periodic resonators or resting on equally spaced supports is investigated by an exact Plane Wave Expansion (PWE) method. This method was first proposed for classical Euler-Bernoulli beams equipped with periodic resonators in ref. [4] and is extended here to stress-driven nonlocal Timoshenko beams. Further, an exact homogeneization approach, which gives an acceptable estimate of opening frequency and size of the band gaps, is also introduced for beams equipped with periodic resonators. Finally, a two-field finite element approach, involving both displacements and internal forces as primary variables, is developed to evaluate the transmittance of finite periodic beams.