To engage means to participate actively in something or to become involved with it. When you engage in a conversation, you listen carefully, think about what's being said, and respond with your own ideas. When a teacher asks students to engage with a challenging math problem, she wants them to really wrestle with it, trying different approaches and thinking it through completely.

The word suggests genuine participation, not just going through the motions. A student might attend class but not truly engage with the lesson, meaning their mind is elsewhere. Someone who engages with a book thinks about the characters, wonders what might happen next, and notices how the author creates suspense or humor.

You can also engage someone in conversation by starting to talk with them, or engage the services of an expert when you hire them to help with something specific. In older usage, when two people became engaged, it meant they promised to marry each other. They were committing to a serious involvement in each other's lives.

The opposite of engagement is detachment or indifference. When you truly engage with something, whether it's a hobby, a friendship, or learning something new, you bring your full attention and energy to it.