In this thesis, we study challenges arising in cosmological data analysis using data from the cosmic microwave background (CMB), remnant radiation from the Big Bang, and the Lyman-α forest, absorption features in spectra of distant quasars. A six-parameter standard model of cosmology, the ΛCDM model, can explain to great accuracy how the Universe evolved from a hot, dense state to the web of galaxies that we observe today. However, it leaves fundamental questions about the nature of dark matter and dark energy unanswered, despite these making up 95% of the observable Universe. The advent of large cosmological surveys present a unique opportunity to infer some of the fundamental laws governing our Universe. Extracting the full potential of this data set is an ongoing challenge because of its size and highly non-linear nature.

In the first part, we present an end-to-end analysis pipeline for large-angular scale CMB data. We present novel foreground removal techniques, improved modeling of the noise and systematics in the data, and develop and extensively test novel likelihood-approximations. The accurate representation of likelihoods including systematics is challenging: exact likelihoods are either unknown or intractable. We present methods that show how to make reliable inference for the optical depth to reionization (τ) or primordial gravitational waves, parametrised by the tensor-to-scalar ratio (r), from large-angular scale CMB data from the Planck satellite. The methods presented range from exact pixel-based likelihoods, maximum-entropy-based semi-analytic likelihood-approximations to simulation-based, so-called likelihood-free, approaches to constrain cosmological parameters. We exhaust current CMB data sets with the developed methods and discuss their potential for next-generation surveys of the CMB.

The upcoming Dark Energy Spectroscopic Instrument (DESI) survey will measure spectra for tens of millions of galaxies and quasars, constructing a three-dimensional map spanning the nearby Universe to 11 billion light years. The Lyman-α forest consists of a series of absorption lines that map the distribution of neutral hydrogen in the intergalactic medium. This allows us to probe the matter distribution of the Universe at intermediate redshifts 2 ≤ z ≤ 5. In the second part of this thesis, we present and develop new continuum fitting methods to extract the un-absorbed flux of a quasar spectrum. We will discuss methods to constrain cosmology using this rich new data set.