In this paper the definition and analysis of the quasimonotone numerical schemes is extended to the general case d > 1, where d is the number of space variables. These schemes are constructed using the simple but very important class of monotone schemes that are defined by two-point monotone fluxes. To enforce the compactness in L∞(Lloc1) of the sequence of approximate solutions, the case of meshes that are a Cartesian product of one-dimensional partitions is addressed. It is proved that the main stability and convergence results for one-dimensional quasimonotone schemes (of the first type) also hold in the general case. As a by-product of this theory, the theory of relaxed, quasimonotone schemes is developed. These schemes are L∞-stable, and they can be more accurate than the quasimonotone schemes; unfortunately, the compactness in L∞(Lloc1 is lost. Nevertheless, if they converge, they do so to the entropy solution.