Complex systems based on nanomaterials and common solvents have been shown to have thermophysical properties that can revolutionize current utilization of heat transfer fluids and heat storage cycles. This has been made possible by the existence of thermal conductivity enhancements derived from the presence of additional mechanisms of heat transfer in comparison with the base solvent. Ionic liquids have been shown to have thermophysical properties that justify the replacement of several of the chemical processes now under exploitation, and some of the solvents used, because they can in certain conditions, be considered as green solvents. Dissolving (or mixing as a thermally stable suspension) nanoparticles in ionic liquids, forms "bucky gels", or IoNanoFluids, which we have recently shown to have thermal conductivity enhancements ranging from (5 to 35) %. This paper reports data on the thermal conductivity of the ionic liquids 1-hexyl-3-methylimidazolium tetrafluoroborate (CAS Number, 244193-50-8), [C(6)mim][BF(4)], 1-butyl-3-methylimidazolium hexafluorophosphate (CAS Number, 174501-64-5), [C(4)mim][PF(6)], 1-hexyl-3-methylimidazolium hexafluorophosphate (CAS Number, 304680-35-1), [C(6)mim][PF(6)], 1-butyl-3-methylimidazolium trifluoromethanesulfonate (CAS Number, 174899-66-2), [C(4)mim][CF(3)SO(3)], and 1-butyl-1-methylpyrrolidinium bis{(trifluoromethyl)sulfonyl}imide (CAS Number, 223437-11-4), [C(4)mpyrr][(CF(3)SO(2))(2)N], and IoNanofluids with multiwalled carbon nanotubes (MWCNTs) as a function of temperature and discuss the molecular theories of heat transfer and storage in these types of systems. Moderate thermal conductivity enhancements, between (2 and 9) %, were found for the systems Studied, showing a week dependence oil temperature. it also reports heat capacity values for [C(4)mim][BF(4)] and [C(4)mim][PF(6)]. Values of the heat capacity of an IoNanofluid, C(4)mim][PF(6)] with (1 and 1.5) % Baytubes, are reported for the first time, showing also an enhancement (8 %), a fact that deserves further investigation in a near future. The behavior of these nanofluids, along with that of ionic liquids of the type studied, Suggests that nanocluster formation and preferred paths for heat transfer and storage are present and are likely to be the cause of the phenomena found. However, existing theories cannot yet explain the results obtained.