noun

• (countable) An appointment, especially to speak or perform.
  "The lecturer has three speaking engagements this week."
• (countable, uncountable) Connection or attachment.
  "Check the gears for full engagement before turning the handle."
• (uncountable, by extension, about human emotional state) The feeling of being compelled, drawn in, connected to what is happening, interested in what will happen next.
  "In a recent study with a movie recommendation site […], higher message interactivity in the form of footprints of user actions, responsive suggestions in a search box, and live-chatting with an online agent led to greater perceived contingency and engagement with the site, which ultimately created more positive attitudes toward it and higher intention to recommend the site to others."
• (countable, uncountable) The period of time when marriage is planned or promised.
  "We are enjoying a long engagement, but haven't yet set a date."