Recent spectroscopic observations of transiting exoplanets have permitted the derivation of the thermal structure and molecular abundances of H₂O, CO, CO₂, CH₄, metallic oxides and alkali metals in these extreme atmospheres. Here, for the first time, a fully-fledged retrieval algorithm has been applied to exoplanet spectra to determine the thermal structure and composition. The development of a suite of radiative transfer and retrieval tools for exoplanet atmospheres is described, building upon an optimal estimation retrieval algorithm extensively used in solar system studies. Firstly, the collection of molecular line lists and the pre-tabulation of the absorption coefficients (k-distribution tables) for high temperature application are discussed. Secondly, the best-fit spectra for hot Jupiters are demonstrated and discussed case by case. Available sets of primary and secondary transit observations of exoplanets are used to retrieve atmospheric properties from these spectra, quantifying the limits of our knowledge of exoplanetary atmospheres based on the current quality of the data. The contribution functions and the vertical sensitivity to the molecules are fully utilised to interpret these spectra, probing the structure and composition of the atmosphere. Finally, the retrievals provide our best estimates of the thermal and compositional structure to date, using the covariance matrices to properly assess the degeneracy between different parameters and the uncertainties on derived quantities for the first time. This sheds light on the range of diverse interpretations offered by other authors so far, and allows us to scrutinise further atmospheric features by maximising the capability of the current retrieval algorithm and to demonstrate the need for broadband spectroscopy from future missions.