This article discusses the behavior of submonolayer quantum films (He and H2) on graphene and newly discovered surfaces that are derived from graphene. Among these substrates are graphane (abbreviated GH), which has an H atom bonded to each C atom, and fluorographene (GF). The subject is introduced by describing the related problem of monolayer films on graphite. For that case, extensive experimental and theoretical investigations have revealed that the phase diagrams of the Bose gases (4)He and para-H2 are qualitatively similar, differing primarily in a higher characteristic temperature scale for H2 than for He. The phase behavior of these films on one side of pristine graphene, or both sides of free-standing graphene, is expected to be similar to that on graphite. We point out the possibility of novel phenomena in adsorption on graphene related to the large flexibility of the graphene sheet, to the non-negligible interaction between atoms adsorbed on opposite sides of the sheet and to the perturbation effect of the adsorbed layer on the Dirac electrons. In contrast, the behavior predicted on GF and GH surfaces is very different from that on graphite, a result of the different corrugation, i.e., the lateral variation of the potential experienced by these gases. This arises because on GF, for example, half of the F atoms are located above the C plane while the other half are below this plane. Hence, the He and H2 gases experience very different potentials from those on graphite or graphene. As a result of this novel geometry and potential, distinct properties are observed. For example, the (4)He film's ground state on graphite is a two-dimensional (2D) crystal commensurate with the substrate, the famous [Formula: see text] phase; on GF and GH, instead, it is predicted to be an anisotropic superfluid. On GF the anisotropy is so extreme that the roton excitations are very anisotropic, as if the bosons are moving in a multiconnected space along the bonds of a honeycomb lattice. Such a novel phase has not been predicted or observed previously on any substrate. Also, in the case of (3)He the film's ground state is a fluid, thus offering the possibility of studying an anisotropic Fermi fluid with a tunable density. The exotic properties expected for these films are discussed along with proposed experimental tests.