Ultracold molecules can be used for a diverse set of exciting applications including controlled quantum chemistry, probes of physics beyond the standard model, simulation of many-body quantum systems and quantum information processing. However their rich internal structure makes the required control of their motional and internal states difficult. In the last decade, the field has made rapid progress on developing techniques for producing, cooling and trapping molecules and, whilst challenges remain, many of these applications are now being realised.

This thesis presents my work on laser cooling and coherent control of calcium monofluoride (CaF) molecules. We load molecules into a three-dimensional magneto-optical trap (MOT), the first such trap for CaF and the second for any molecule. The MOT contains around 2×10⁴ molecules with an initial temperature of 12 mK. Lowering the laser intensity brings the temperature of the MOT down to 960 μK. To further cool the molecules we load them into a blue-detuned optical molasses, cooling them to 5.4(7) μK. This temperature is ~40 times below the Doppler limit and corresponds to a typical molecule moving at only a few times the single-photon recoil velocity. We optically pump the molecules into a single Zeeman sub-level and coherently transfer them between rotational states using microwave radiation. We load the molecules into a magnetic trap where they remain confined for about 5 s, the lifetime being limited by vibrational excitation due to room-temperature blackbody radiation. We show how to prepare the trapped molecules in a coherent superposition of two rotational states, and how to maintain this coherence for long periods, up to 6.4(8) ms. Detailed simulations of the remaining decoherence mechanisms suggest that it is possible to extend this to more than a second by compressing the sample to a smaller size, and using a magnetic trap geometry where the direction of the field is nearly uniform across the trapping volume. Finally we explore theoretically how Raman sideband cooling can be applied to molecules in optical tweezer traps, potentially providing a route to full quantum control.