Recent theoretical and numerical models for the motion of saltating particles close to stream beds are constituted of three sub-models: a) a set of equations describing the particle "free flight", b) a sub-model to calculate the post-collision particle velocity, and c) a mathematical representation of the bed roughness. In this paper, a comprehensive three-dimensional (3-D), theoretical/numerical model for bed-load motion at large Reynolds numbers is presented. By using geometric considerations and stochastic parameters to characterize collisions with the wall, five new sub-models for representation of bed roughness are, for the first time to the best of our knowledge, proposed and implemented. The emphasis of this paper is on the particle model, for which Basset, Magnus, drag, submerged weight, virtual mass, and lift forces are included. For the range of particle sizes (sands) analyzed herein, it is found that the stream-wise contribution of the Basset force, compared to other forces, may be as large as 60%. Whereas in the wall-normal direction, the Basset force is equally important as the drag force, and it is exceeded only by the submerged weight. It is also found that the best agreement between numerical and experimental results in terms of jump length, jump height, and stream-wise particle velocity is achieved for restitution and friction coefficients of 0.65 and 0.1, respectively. Important conclusions are obtained regarding the lack of realistic prediction with available "roughness" models with small ranges of angles. (C) 2019 International Research and Training Centre on Erosion and Sedimentation/the World Association for Sedimentation and Erosion Research. Published by Elsevier B.V. All rights reserved.