Modern gas turbine engines run at extremely high temperatures which require the high pressure turbine blades to be extensively cooled in order to reach life requirements. This must be done using the minimum amount of coolant in order to reduce the negative impacts on the cycle efficiency. In the design process the cooling configuration and stress distribution must be carefully considered before verification of the design is conducted. Improvements to all three of these blade design areas are presented in this thesis which investigates internal cooling systems in the form of ribbed, radial passages and leading edge impingement systems. The effect of rotation on the heat transfer distribution in ribbed radial passages is investigated. An engine representative triple-pass serpentine passage, typical of a gas turbine mid-chord HP blade passage, is simulated using common industrial RANS CFD methodology with the results compared to those from the RHTR, a rotating experimental facility. The simulations are found to perform well under stationary conditions with the rotational cases proving more challenging. Further study and simulations of radial passages are undertaken in order to understand the salient flow and heat transfer features found, namely the inlet velocity profile and rib orientation relative to the mainstream flow. A consistent rib direction gives improved heat transfer characteristics whilst careful design of inlet conditions could give an optimised heat transfer distribution. The effect of rotation on the heat transfer distribution in leading edge impingement systems is investigated. As for the radial passages, RANS CFD simulations are compared and validated against experimental data from a rotating heat transfer rig. The simulations provide accurate average heat transfer levels under stationary and rotating conditions. The full target surface heat transfer in an engine realistic leading edge impingement system is investigated. Experimental data is compared to RANS CFD simulations. Experimental results are in line with previous studies and the simulations provide reasonable heat transfer predictions. A new method of combined thermal and mechanical analysis is presented and validated for a leading edge impingement system. Conjugate CFD simulations are used to provide a metal temperature distribution for a mechanical analysis. The effect of changes to the geometry and temperature profile on stress levels are studied and methods to improve blade stress levels are presented. The thermal FEA model is used to quantify the effect of HTC alterations on different surfaces within a leading edge impingement system, in terms of both temperature and stress distributions. These are then used to provide improved target HTC distributions in order to increase blade life. A new method using Gaussian process regression for thermal matching is presented and validated for a leading edge impingement case. A simplified model is matched to a full conjugate CFD solution to test the method's quality and reliability. It is then applied to two real engine blades and matched to data from thermal paint tests. The matches obtained are very close, well within experimental accuracy levels, and offer consistency and speed improvements over current methodologies.