The cosmic microwave background (CMB) is made up of photons, the majority of which last scattered with electrons during the era of cosmic recombination, at redshift z~1100. They have been travelling, virtually undisturbed, ever since. Along their journey, however, the photons' paths are deflected by the gravitational influence of the matter distribution of the Universe --- an effect known as `gravitational lensing'. Though the deflections are small, they can affect the statistical properties of the ensemble in crucial ways.

Lensing converts part of the E-mode polarisation of the CMB into B-modes, thus introducing a source of noise when searching for a primordial B-mode signal associated with primordial gravitational waves (PGWs) generated during cosmic inflation. Fortunately, constraints on PWGs can be improved by partially removing lensing B-modes --- i.e., `delensing' them.

The core of this thesis consists of work towards an improved understanding of the process of B-mode delensing. We dedicate a chapter to exploring the benefits and limitations of delensing B-modes using templates constructed by combining high-resolution measurements of E-modes with proxies of the matter distribution responsible for the lensing deflections. We prove the counter-intuitive result that a gradient-order template is more effective when constructed from lensed E-modes, instead of their delensed or unlensed version. Furthermore, we show that, given lensed E-modes, a gradient-order template is to be preferred over a non-perturbative one. A gradient-order template built from lensed E-modes will be effectively optimal for all planned CMB experiments, with the added benefit of being analytically transparent and computationally efficient.

In another chapter, we consider the case where B-modes are delensed using external tracers of the matter distribution of the Universe. We discuss how uncertainties in measurements of the tracer spectra might translate to biased constraints on PGWs, and show that this effect will be negligible for the upcoming Simons Observatory (SO). Then, we focus on the problem of foreground residuals when the cosmic infrared background (CIB) is used to delens B-modes. Significant biases can arise if the largest angular scales are included in analyses, because of non-Gaussian galactic dust residuals remaining after foreground cleaning. However, we propose mitigation techniques, and show that they will be highly effective.

We also revisit internal delensing biases arising when B-modes are delensed using a lensing reconstruction obtained from a quadratic combination of E- and B-modes. We learn that these biases necessarily lead to degraded constraints on PGWs, despite appearing to reduce the variance of the estimators we use to constrain them. We derive an analytic model for the bias, but show that it is generally advantageous to exclude the overlapping modes rather than model or renormalise the bias.

In the final original chapter, we address the issue of extragalactic foreground contamination to temperature-based lensing reconstructions. We present an analytic framework for calculating the biases associated with the thermal Sunyaev-Zeldovich effect and the CIB using the halo model. Our predictions can be evaluated very efficiently (at least for the dominant set of terms) and are in reasonable agreement with simulations.