A novel movable-element enabled simultaneously transmitting and reflecting surface (ME-STARS) communication system is proposed, where ME-STARS elements positions can be adjusted to enhance the degress-of-freedom for transmission and reflection. For each ME-STARS operating protocols, namely energy-splitting (ES), mode switching (MS), and time switching (TS), a weighted sum rate (WSR) maximization problem is formulated to jointly optimize the active beamforming at the base station (BS) as well as the elements positions and passive beamforming at the ME-STARS. An alternative optimization (AO)-based iterative algorithm is developed to decompose the original non-convex problem into three subproblems. Specifically, the gradient descent algorithm is employed for solving the ME-STARS element position optimization subproblem, and the weighted minimum mean square error and the successive convex approximation methods are invoked for solving the active and passive beamforming subproblems, respectively. It is further demonstrated that the proposed AO algorithm for ES can be extended to solve the problems for MS and TS. Numerical results unveil that: 1) the ME-STARS can significantly improve the WSR compared to the STARS with fixed position elements and the conventional reconfigurable intelligent surface with movable elements, thanks to the extra spatial-domain diversity and the higher flexibility in beamforming; and 2) the performance gain of ME-STARS is significant in the scenarios with larger number of users or more scatterers.