Universal relations for dilute, strongly interacting Fermi systems can be studied in ultracold fermionic 6Li quantum gases using Bragg spectroscopy. Standard techniques of laser trapping and cooling of atoms are applied to evaporate fermionic gases below the critical temperature for superfluidity, TC. The atoms are prepared to equally populate the lowest two spin states and the interaction between atoms of opposite spin is precisely tunable via a Feshbach resonance. A unitary Fermi gas represents a special case where the interactions are categorised as strong, that is, the range of interaction exceeds the range of the interatomic potential. Elastic collisions in the gas become unitarity limited and independent of details of the atomic properties. This universality is a property of unitary Fermi gases and the universal relations adequately describe the thermodynamic behaviour of these gases. The universal contact parameter is a central quantity that encapsulates the microscopic details of the system and varies with the temperature and interaction strength. The principal topic of this thesis is the use of Bragg spectroscopy as a tool to measure the universal contact in trapped Fermi gases. The contact parameter quantifies the short-range pair correlation function according to a universal law. Inelastic Bragg scattering of photons allows one to measure the static structure factor, which is by definition the Fourier transform of the pair correlation function. Due to this well defined relation between the contact and Bragg spectra, the universal relation for the static structure factor is experimentally verified for a range of transferred momenta, k/kF = 3.5 − 9.1, in agreement with the theoretical predictions. The contact in the zero-temperature limit at different interaction strengths as well as in the unitarity limit at different temperatures agrees well with the calculations. The contact at unitarity decreases monotonically for T/TF = 0 − 1 but has a non-zero value well above the critical temperature TC ≈ 0.2 TF . The contact parameter may offer new insights into criticality or complex superfluid pairing mechanisms such as pseudogap-pairing.