This chapter investigates stability and localizability of local-bearing-based multi-agent systems without common orientation in the presence of noisy measurement, which are more general but also more challenging to deal with than global-bearing-based multi-agent systems under ideal environment. Based on local-bearing unbiased estimator constructed from the historical information and a newly designed time-varying gain, a new robust prescribed-time orientation estimation algorithm is proposed to ensure that the local reference frame of the follower agent is aligned with the global one. The local bearing information is more easily obtained than global one. Therefore, the new orientation estimation result is expected to be more widely applicable. The robust orientation estimation algorithm is then applied to the problem of localization estimation, and a prescribed-time distributed localization estimation algorithm is developed. The distinctive advantage of this work is that only the local bearing information is used, and the fast and controllable localization estimation is achieved. The global convergence is derived based on the cascade system and some simulation results are provided to prove the effectiveness of the proposed estimation algorithms.