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The polyphagous tropical Mediterranean fruit fly (Ceratitis capitata Weid. (medfly)) was detected in California in 1975, and a large-scale detection/eradication campaign was begun in the absence of sound knowledge of the fly’s potential invasiveness and geographic distribution. Persistent measurable populations of the fly have not been found in California, but a scientific explanation for this has not developed. A physiologically based demographic system model (CASAS) was developed to examine the effects of temperature on medfly’s potential distribution across the ecological zones of Arizona-California (AZ-CA), and in Italy where the fly is established. The system model simulates the daily age-mass structured dynamics of a tree host composed of sub-unit populations of leaves, stem, roots and fruit, as well as the age-structured dynamics of medfly life stages. Total pupae tree(-1) year(-1) was used as the metric of favorability for medfly at 151 locations in AZ-CA during 1995-2006, and at 84 locations in Italy during 1999-2005. The results were mapped using GRASS GIS. AZ and the southern desert areas of CA are unfavorable for medfly because of high summer temperatures, while much of CA, including many frost-free areas, is too cold. Only the area of south coastal CA (San Diego, Orange and Los Angeles Counties) is potentially favorable for medfly, but in the absence of measurable populations, we cannot say whether it is established there. The majority of medfly discoveries over the past 35 years have occurred in south coastal CA, but discoveries also occurred in Santa Clara County in northern CA, mostly during 1975 and 1980-1981. Santa Clara County, just south of San Francisco Bay, is generally marginal for medfly, but favorability increased approximately 10% during the period 1979-1981. Medfly has been established in Italy for decades, and our model predicts its wide distribution in the southern and western regions of the country. The fly is restricted in northern areas and at higher elevations of Italy by winter temperatures. Temperature is expected to increase in CA and the Mediterranean Basin. We used two scenarios consisting of increasing observed daily temperatures +2 and +3 degrees C to examine the effects on the potential distribution of the fly in CA and Italy. Increasing temperatures expand the favorable range for medfly northward along the coast of CA, but decrease it in the southern reaches of current favorability. A similar but greater increase in geographic range is predicted for Italy. We examine critically some ongoing eradication programs in CA, and question the scientific basis for them. We also review some climate matching approaches used to assess the potential geographic distribution of invasive species.