Each cell in the grid can be addressed by index (i, j) in two dimensions or (i, j, k) in three dimensions, and each vertex has coordinates in 2D or in 3D for some real numbers dx, dy, and dz representing the grid spacing.
A rectilinear grid is a tessellation by rectangles or parallelepipeds that are not, in general, all congruent to each other. The cells may still be indexed by integers as above, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a rectilinear grid that is not regular appears on logarithmic scalegraph paper.
A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which the cells are quadrilaterals or cuboids rather than rectangles or rectangular parallelepipeds.