This is a review of petroleum asphaltene solubility as it relates to production and refining in terms of changes in temperature, pressure, and composition. While the current concept of asphaltenes and asphaltene solubility is changing rapidly in the scientific literature, giving the appearance of a field in chaos, evidence is presented that the many of the older approximations may still be adequate enough to solve practical asphaltene insolubility problems. Actually, solubility parameters and the regular Flory-Huggins model may describe petroleum phase behavior better than they do for any other system. The oil compatibility model for avoiding asphaltene insolubility during the processing of crude oils is discussed in detail. For production applications with live oils at reservoir conditions, the objective is to predict asphaltene solubility or insolubility from only measurements at ambient conditions and knowledge of the quantity and composition of the dissolved gas. Several groups have reported achieving this goal using either the regular Flory-Huggins model or the perturbed chain modification of the statistical associating fluid theory (PC-SAFT) equation of state.