The present study is focused on the simulation of turbulent bubbly flows by utilizing the two-fluid model (TFM) in conjunction with advanced near-wall Reynolds-stress models (RSMs) within the Reynolds-averaged Navier-Stokes (RANS) framework. Such anisotropy-resolving turbulence models, employed in combination with the TFM, have been rarely used so far for two-phase flow computations. The presently adopted RSMs are based on the formulations initially proposed by Jakirlic and co-workers for incompressible single-phase flows. Two essentially different RSM versions are selected to be applied in the present work. One model version is formulated within the conventional RANS framework, whereas the second one resembles an instability-sensitized RSM variant, capable of adequately resolving the fluctuating turbulent motions in accordance with the scale-adaptive simulation (SAS) proposal by Menter and Egorov. The necessary modifications of both Reynolds-stress models to be used within the TFM computational framework, also in conjunction with different model formulations accounting for the bubble-induced turbulence, require an appropriate coupling algorithm, which, independent of the geometrical complexity of the flow configurations considered, represents by itself a challenging task. The three reference flow configurations, chosen for the model validation, are the turbulent bubbly flows in a straight and suddenly-expanded vertical pipe over a range of Reynolds numbers and a square cross-sectioned bubble column. In addition, due to sake of comparative evaluation, the available corresponding single-phase flows are investigated by using both RSMs. The presently realized numerical investigations demonstrate a successful employment of both Reynolds-stress models for bubbly flow computations. In all three flow configurations the results obtained with the conventional RSM exhibit a high level of qualitative and quantitative agreement with the available reference data. The novel scale-resolving RSM reveals its high potential, representing a promising approach for further investigations.