Different families of fullerenes taking as a basis their progenitors, mini-fullerenes, and graph theory were constructed. They are: a family of a bi-trefoil: C14, C18, C24, C30, C36; a family of truncated bipyramids: C14, C18, C24 C30, C36; a family of cupola half-fullerenes: C10, C12, C16, C20, C24; and a family of 4-6-equator fullerenes: C20, C24, C32, C40, C48. All the families contain five members and have a layer structure. By analogy with geography, we can distinguish two frigid zones composed of pentagons and a torrid zone composed of hexagons. The smallest fullerenes of each family are exceptions; its frigid zones are formed of squares. Another interesting feature is that the torrid zone in the family of 4-6-equator fullerenes: C20, C24, C32, C40, and C48 is composed of square-hexagon pairs. We have also constructed a family of cupola half-fullerenes: C10, C12, C16, C20, and C24 with the help of transition to modified graphs introduced in , where a cluster of three or four atoms being considered as a big point (vertex) contrary to a zero-size point (vertex) of a common graph.