When numerically integrating the equation describing the motion of a particle in a carrier fluid, the computation of the Basset (history) force becomes by far the most expensive and cumbersome, as opposed to forces such as drag, virtual mass, lift, buoyancy and Magnus. The expression representing the Basset force constitutes an integro-differential term whose standard integrand is singular when the upper integration limit is enforced. These shortcomings have led some researchers to either disregard or outright neglect the contribution of the Basset force to the total force, even in those cases where it may yield to important errors in the determination of particle trajectories in the computation of sediment transport and other environmental flows. This work is devoted to review four recent contributions associated with the computation of the Basset force, and to compare their proposals to diminish the inherent problems of the term integration. All papers, except one, use variants of a window-based approach; the most recent contribution, in turn, employs a specialized quadrature to increase the accuracy of the computation. An analysis was carried out to compare CPU computation times, rates of convergence and accuracy of the approximations versus a known analytical solution. All methods provide sound solutions to the issues associated with the computation of the Basset force; further, a road map to select the best solution for each given problem is provided. Finally, we discuss the implications of the techniques for the simulation of sediment transport processes and other environmental flows.