This is an experimental and theoretical study of the slow translation of a hot sphere through a fluid at rest at infinity. The viscosity depends strongly on temperature, i.e., if ΔT = T0 − T∞ is the applied temperature difference and γ = |(d/dT0) lnμ(T0)|, then the parameter θ = γ ΔT is large: it is about 6.5 in the experiments and is taken as infinite in the theory. The flow is determined by two large parameters, namely the Nusselt number N and the modified viscosity ratio ε−1 = ν∞/(ν0θ3). The qualitative state of the flow is observed to depend on the relation between N and ε. If ε−1 → ∞ (N fixed, possibly large) previous analysis (Morris 1982) shows that all the shear occurs in a thin low-viscosity film coating the sphere; this film and the associated thermal layer separate at the equator, and a separation bubble of low-viscosity fluid trails the sphere. (ii) If N → ∞ (ε−1) large but fixed) even the most viscous fluid deforms, and both the drag and heat losses are found to be controlled by this highly viscous flow. The present work maps the major asymptotic states which separate these two end-states for small ε. The drag and heat-transfer laws are determined experimentally and theoretically: in addition it is shown that separation of the thermal layer ceases when the drag is controlled by the most viscous fluid, even though the heat transfer in this case can be still controlled by the dynamics of the least-viscous fluid. The heat-transfer and drag laws are also given for a sphere moving in a spherical container of finite radius. This model is shown to give a close estimate of wall effects for a sphere moving in a cylindrical container. For state (i) the theory predicts the heat transfer to within 20% and, for the smallest ε, the drag to within 30%. In the experiments ε is small enough for all limiting states to be evident but, apart from state (i), a design flaw prevents a quantitative test of the theory. For the other states, the theory is compared with numerical results from Daly & Raefsky (1985). Although the values of ε in the calculations are not small enough for the limiting states to be achieved, the theory predicts the drag to within 8% and the heat transfer to within 10 %.