Heavy oils are complex fluids and their flow properties are of primary importance to the assessment of their commercial value or to the design of production and transport facilities. Compounds such as asphaltenes and wax crystals, for instance, are known for their complex physical behaviors and interactions, and consequently they highly contribute to the macroscopic flow behavior of the crude oil. In this study, we investigate two particular aspects of the heavy oil flow behavior: the temperature dependence of the viscosity and the rheological and structural properties. The viscosity and viscoelasticity of a set of 13 different natural heavy oils from various origins (Asia and North, Central, and South America) are characterized over a wide range of temperatures. The zero-shear viscosity is measured from -40 degrees C to 200 degrees C, and the data are interpreted through the concept of glass transition, experimentally observed by differential scanning calorimetry (DSC) and fitted by the Williams Landel-Ferry (WLF) model. The fragility of the different oils is found to be very similar throughout the sample set and the WLF constants are similar to the universal values observed in polymers. A detailed rheological characterization of the oils is also undertaken, under steady-shear experiments and dynamic oscillatory tests at temperatures from -50 degrees C to 50 degrees C. Independently of their zero-shear viscosities, the heavy oils have different rheological properties ranging from a Newtonian and purely viscous character to a weak gel-like behavior linked to some elastic internal structure. The viscoelasticity is quantified through the relaxation exponent n, which is then matched to some compositional features. For some oils, the viscoelastic character is linked to the presence of paraffinic wax crystals, the amount of which is quantified by DSC. For the other viscoelastic oils, the elastic character seems to be related to their high amount of asphaltenes: there is indeed a trend between the asphaltene content and the relaxation exponent, suggesting that the asphaltenes, when present in high quantities, are linked to the structural elastic properties, which lead to the macroscopic weak gel-like behavior.