The interaction of turbulence with shock waves, while very common in nature and engineering systems, is a very difficult problem from a theoretical, numerical and experimental perspective. A main challenge arise from the two-way coupling between the shock and turbulence which occurs at a wide range of scales in time and space. The focus of this work is on the fundamental understanding of these shock-turbulence interactions (STI) at high turbulence intensities with high-fidelity direct numerical simulations (DNS) that fully resolve the shock. The numerical study is guided by novel theoretical work that results in analytical expressions for thermodynamic jumps across the shock which depend on turbulence characteristics. Comparison with DNS data shows that these expressions can indeed predict quantitatively a number of statistical variables of interest. The theory presented here also predicts distinct shock amplification for the first time. Results on the shock structure are used to validate previous theoretical proposals and extend the analysis to much stronger interactions which leads to the observation of a new regime (vanished shocks) in which turbulence undergoes a classical spatial decay as it across the shock. Finally, the amplification of turbulence across the shock is discussed. Disagreements in the literature on Reynolds stresses are solved by recognizing a special kind of similarity scaling on two different parameters in two different limits.