What can cause severe pain in left side up and under my ribs?

Many possibilities: Chest pain with deep breaths can be due to a number of causes, including costochondritis (a rib/cartilage pain, a rib fracture, muscular strain, a pneumothorax (partially collapsed lung), any number of lung diseases, and even heart disease. Although costochondritis is common, be sure to see your doctor to rule out other more serious causes, or the er if severe pain or short of breath.