To date the LHC has not found evidence for the existence of the new states that are predicted to exist if supersymmetry (SUSY) provides a solution to the hierarchy problem. Moreover a Higgs scalar mass of O(126 GeV), as indicated by preliminary measurements at the LHC, lies above the preferred range of the simplest SUSY theory (MSSM). However SUSY must be broken and the non-observation of SUSY states may indicate that there are strong correlations between the soft SUSY breaking terms that must be added to the theory. Theoretically-well-motivated correlations can lead to a significant increase in the expected mass of the SUSY states without requiring significant fine-tuning. The Higgs mass can also be raised in extensions of the MSSM. I will discuss the structure of these extensions and prospect for testing them at the LHC, both in the supersymmetric and Higgs scalar sectors, and in dark matter searches.