The traditional approach to the stress testing of financial institutions focuses on capital adequacy and solvency. Liquidity stress tests are often applied in parallel to solvency stress tests, based on scenarios which may not be consistent with those used in solvency stress tests. We propose a structural framework for the joint stress testing of solvency and liquidity: our approach exploits the mechanisms underlying the solvency-liquidity nexus to derive relations between solvency shocks and liquidity shocks. These relations are then used to model liquidity and solvency risk in a coherent framework, involving external shocks to solvency and endogenous liquidity shocks. We introduce solvency-liquidity diagrams as a method for analysing the resilience of a balance sheet to the resulting combination of solvency shocks and endogenous liquidity shocks. Finally, we define the concept of 'Liquidity at Risk' which quantifies the liquidity resources required for a financial institution facing a stress scenario.