You can build a bowl with a lathe. You can build one with a special router bit. Heck, you can use wood carving tools to hollow out a blank.
But, did you realize you could cut one with a scroll saw?
Oh, yes, and you can do it and make it look pretty darned sweet, too. This page on the Instructables website shows the process form stock preparation to blade selection to cutting process.
While these basic instructions show you how to build a bowl of a certain dimension, they guide you on how to calculate the cutting angles if you want to create a bowl of your own design.