The original definition of the IFS with affine contractive mappings is an important and handy tool for constructive approach to fractal sets. But, in spite of clear definition, the concept of IFS does not allow many possibilities in the sense of modeling of such sets, typically being fairly complicated. One step in direction of improving the concept of IFS consists in introducing AIFS, a variant of IFS that permits affine invariance property which is vital from the point of modeling. The theory is supported by comprehensive examples.