1. General: latitude or power, but not an obligation, to exercise a right or to do or not do something. 
2. Law: right of a corporate or public officer (such as a judge) to act according to his or her judgment or conscience in certain circumstances but within given limits and under the guidance of customary principles. 
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USAGE EXAMPLE

The boss told her employees who attended the meeting to use their discretion when evaluating expense requests and to reject any requests that did not seem reasonable.