We analyse the large and diverse literature on technical change in integrated assessment models (IAMs) of climate change, with a view to understanding how different representations of technical change affect optimal climate policy. We first solve an analytical IAM that features several models of technical change from the literature, including exogenous technical change in abatement technologies, exogenous decarbonisation of the economy, endogenous technical change via learning-by-doing, and endogenous technical change via R&D (in particular, directed technical change). We show how these models of technical change impact optimal carbon prices, emissions and temperatures in often quite different ways. We then survey how technical change is currently represented in the main quantitative IAMs used to inform policy, demonstrating that a range of approaches are used. Exogenous technical change in abatement technologies and learning-by-doing are most popular, although the latter mechanism is only partially endogenous in some models. We go on to quantify technical change in these policy models using structural estimation, and simulate our analytical IAM numerically assessing the effect of technical change on optimal climate policy. We find large quantitative effects of technical change and large quantitative differences between different representations of technical change, both under cost-benefit and cost-effectiveness objectives.