The strength of the magnetic field is one of the fundamental properties of the Earth, and its behaviour over time has implications in disparate fields such as geodynamics and archaeology. Thermal remanent magnetization (TRM) has a quasi-linear relationship to the ambient magnetic field applied during cooling of magma. This process can be reproduced in the laboratory, making it possible to estimate absolute paleointensity of Earth's magnetic field. TRM, of all the forms of remanent magnetization formed in nature, has the strongest theoretical basis thanks to the work of Néel (1949) and Thellier & Thellier (1959). Despite the simplicity of TRM theory for ideal, uniformly magnetized grains, there are many complications that make interpretation of paleointensity data difficult. And there are clues in the present data base that things can go very wrong. For example, although we know that paleomagnetic directions on Earth’s surface are well explained by a simple geocentric axial dipole field model, intensity data for even the best studied lava flow (Hawaii, 1960) have estimates spanning the entire range on Earth’s surface and even higher. We must do better! Recent results from micromagnetic modeling, laboratory analogue experiments, and new approaches to data selection and field sampling lead to the optimistic view that accurate estimates are achievable. In this lecture I will review where we are, how we got there and where we can go with paleointensity estimates.