Lattice rule algorithms for the purpose of approximating continuous functions in several variables are studied. Here, the number of unknowns may be very large. The dependence of the function to be approximated on its various components can be weighted by a weight factor, which may mean that the function depends strongly or weakly on that component of its argument. The approximation is fixed by the function values at lattice points. The lattice is generated by an algorithm and depends on the approximand. The tractability of the problem is of special interest, and necessary and sufficient conditions for tractability or strong tractability are derived. Methods for generating the lattice and for generating vectors e.g. for integration are provided, too. Many numerical results illustrate the theory.