The Moon doesn't travel around the world, it travels around the Earth! And it takes approximately 27.3 days for the Moon to complete one orbit around the Earth. This is called a sidereal month.

It's important to note that this is the time it takes for the Moon to return to the same position relative to the stars. The time it takes for the Moon to go through its phases (from new moon to full moon and back again) is slightly longer, about 29.5 days, and is called a synodic month.