Exquisitely sensitive measurements of Cosmic Microwave Background (CMB) fluctuations have made it possible to determine with great precision the Universe's inventory, as well as basic properties of its initial conditions. This required a highly accurate theoretical modeling of the underlying physics, in particular of the cosmological recombination process, which I will describe in the first part of this talk. A wealth of new and complementary data will be collected in the decades to come, promising spectacular advances in our yet very incomplete understanding of the cosmos. In the remainder of this talk, I will highlight some upcoming cosmological probes and illustrate fundamental questions they will help us answer. I will first discuss CMB spectral distortions, a powerful probe the Universe's early thermal history, and illustrate how they can be used to probe, for instance, the properties of the dark matter particle. I will then describe how large-scale structure can be used to measure the yet unknown neutrino mass. I will end by describing the exciting prospects of high-redshift 21-cm cosmology, in particular for characterizing the Universe's initial conditions.