This work is dedicated to Korovkin type theorems in Banach function spaces. The subspace X-S of the Banach function space X generated by the shift operator is considered and the density of the set C-0(infinity) in X-S is proved. The analogs of the Korovkin theorems in X-S are obtained. Also, the analog of the Korovkin theorem for Kantorovich polynomials is derived both in the cases of rearrangement-invariant and general non-rearrangement-invariant Banach function spaces. These results are obtained for Lebesgue spaces, grand-Lebesgue spaces, Morrey-type spaces and their weighted versions, weak Lebesgue spaces, Orlicz spaces. Note that in our case the Korovkin-type theorem for Kantorovich polynomials in Morrey spaces is an only natural analog of the classical L-p version of the Korovkin theorem and strengthens the previously known result.