Recombination is a fundamental physical process that describes how the Universe transformed from a primeval plasma of protons and electrons into neutral atoms and an expanding shell of decoupled photons. This process is integral to the calculation of the CMB anisotropies and includes some fascinating atomic physics as well as room for exotic cosmological models. In this thesis, the recombination calculations have been performed using Recfast++ and CosmoRec, the recombination lines have been performed using CosmoSpec and the CMB anisotropies have been modelled using CAMB. In the first chapter, we provide the background of cosmology as well as a description of the cosmic microwave background (CMB). In Chapter 2, we introduce the recombination problem and the detailed atomic physics that was used to solve it. In Chapter 3, we explore the recombination epoch with variations of fundamental constants. Time-dependent variations of these constants, that stretch and compress the Thomson visibility function, are also shown using a simple power law. We study how these effects alter the CMB anisotropies and present the most precise constraints with the Planck 2015 data using the Markov Chain Monte Carlo code CosmoMC. We compare these results with the more recent 2018 release of the Planck data in the second part of Chapter 3. The geometric degeneracies between the electron mass and Hubble's constant, that are shown for previous Planck datasets as well, opens up even wider in the 2018 release. If the CMB data is coupled with SH0ES data for supernovae and Cepheid variables, we can see a reduction in the Hubble tension due to the electron mass-Hubble degeneracy. In Chapter 4, we summarise the principal component analysis technique for cosmology and present our new code FEARec++. It generates new eigenmodes that are exceptionally orthogonal and efficiently decoupled from standard parameters. These components have been constrained with Planck 2015 data, disfavouring non-standard recombination, and improving the error on the third, worst-constrained eigenmode by ~ 62%. We present a `direct projections' method that can be used as an efficient parameter estimation tool when one needs a quick constraint and has successfully generated orthogonal eigenmodes. Finally, in Chapter 5 we provide a brief description of the cosmological recombination radiation along with the exceptional work that has been done to fulfill this computation with CosmoSpec. This culminates with recent modeling of recombination line variations due to changes in the cosmological parameters. In conclusion, exotic physics and principal components allow us to effectively probe recombination and have led to tools which can be applied to the cosmological recombination radiation.