Your pursuit of something is your attempts at achieving it. If you do something in pursuit of a particular result, you do it in order to achieve that result.

The pursuit of an activity, interest, or plan consists of all the things that you do when you are carrying it out.

Someone who is in pursuit of a person, vehicle, or animal is chasing them.

In cycling and skating, the pursuit is a race in which two competitors or teams start on opposite sides of a circular track and try to catch up with each other.

Your pursuits are your activities, usually activities that you enjoy when you are not working.

If you are in hot pursuit of someone, you are chasing after them with great determination.