Climate change will alter the distribution and abundance of many species, including those of concern to human health. Accurate predictions of these impacts must be based on an understanding of the mechanistic links between climate and organisms, and a consideration of evolutionary responses. Here we use biophysical models of energy and mass transfer to predict climatic impacts on the potential range of the dengue fever vector, Aedes aegypti, in Australia. We develop a first-principles approach to calculate water depth and daily temperature cycles in containers differing in size, catchment and degree of shading to assess habitat suitability for the aquatic life cycle phase. We also develop a method to predict potential climatic impacts on the evolutionary response of traits limiting distribution. Our predictions show strong correspondence with the current and historical distribution and abundance of Ae. aegypti in Australia, suggesting that inland and northern limits are set by water availability and egg desiccation resistance, and southern limits by adult and larval cold tolerance. While we predict that climate change will directly increase habitat suitability throughout much of Australia, the potential indirect impact of changed water storage practices by humans in response to drought may have a greater effect. In northern Australia, we show that evolutionary changes in egg desiccation resistance could potentially increase the chances of establishment in a major centre (Darwin) under climate change. Our study demonstrates how biophysical models of climate-animal interactions can be applied to make decisions about managing biotic responses to climate change. Mechanistic models of the kind we apply here can provide more robust and general predictions than correlative analyses. They can also explicitly incorporate evolutionary responses, the outcomes of which may significantly alter management decisions.