We study the vulnerability of 130 banks directly supervised by the European Central Bank's Single Supervisory Mechanism. Illustrative stress tests using banks' balance sheet data reveal that significant stress prevails in the euro area's smaller and medium-sized banks, many of them located in southern Europe. The banks we identify as stressed also have performed substantially worse on the stock market. The vulnerable banks are typically hobbled by non-performing loans to European businesses. Strengthening the banking system, therefore, is important to achieve sustainable recovery because it will revitalise credit to the healthier segments of the economy. But instead of emphasising bank recapitalisation, as in past years, we believe the task is to shrink the banking sector to a healthier core.