One of the main goals of modern observational cosmology is to constrain or detect a stochastic background of primordial gravitational waves. The existence of such a background is a generic prediction of the inflationary paradigm: the leading explanation for the universe's initial perturbations. A detection of the gravitational wave signal would provide strong evidence for the paradigm and would amount to an indirect probe of an energy scale far beyond that of conventional physics. Several dedicated experiments search for the signal by performing highly accurate measurements of a unique probe of the primordial gravitational wave background: the B-mode signature in the polarization of the cosmic microwave background (CMB) radiation. A part of this thesis is devoted to one of these experiments: the balloon-borne Spider instrument. The analysis of the first dataset, obtained in two (95 and 150 GHz) frequency bands during a January 2015 Antarctic flight, is described, along with details on the characterisation of systematic signal and the calibration of the instrument. The case of systematic signal due to poorly understood optical properties is treated in more detail. In the context of upcoming experiments, a study of systematic optical effects is presented as well as a numerically efficient method to consistently propagate such effects through an analysis pipeline. This is achieved by a `beam convolution' algorithm capable of simulating the contribution from the entire sky, weighted by the optical response, to the instrument's time-ordered data. It is described how the algorithm can be employed to forecast the performance of upcoming CMB experiments. In the final part of the thesis, an additional use of upcoming B-mode data is described. Constraints on the non-Gaussian correlation between the large-angular-scale B-mode field and the CMB temperature or E-mode anisotropies on small angular scales constitute a rigorous consistency check of the inflationary paradigm. An efficient statistical estimation procedure, a generalised bispectrum estimator, is derived and the constraining power of upcoming CMB data is explored.