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1. To pursue someone or something intently and tenaciously. I've always wanted to be a doctor; I've been chasing after that dream my whole life. The security guards chased after the thief for nearly a mile.
2. To pursue or court someone romantically or sexually. Mary has been chasing after Joe ever since she developed a crush on him in high school. My brother always spends his weekends chasing after girls in bars around town.