Infinity captivates the imagination. A child stands between two mirrors and sees herself reflected over and over again, smaller and smaller, trailing off to infinity. Does it go on forever? Does anything go on forever? Does life go on forever? Does time go on forever? Does the universe go on forever? Is there anything that we can be certain goes on forever? It would seem that the counting numbers go on forever, since given any number on can always add one. But is that the extent of forever? Or are there numbers that go beyond that? Are there higher and higher levels of infinity? And, if so, does the totality of all of these levels of infinity itself constitute the highest, most ultimate, level of infinity, the absolutely infinite? In this seminar we will focus on the mathematical infinite. We will start with the so-called "paradoxes of the infinite," paradoxes that have led some to the conclusion that the concept of infinity is incoherent. We will see, however, that what these paradoxes ultimately show is that the infinite is just quite different than the finite and that by being very careful we can sharpen the concept of infinity so that these paradoxes are transformed into surprising discoveries. We will follow the historical development, starting with the work of Cantor at the end of the nineteenth century, and proceeding up to the present. The study of the infinite has blossomed into a beautiful branch of mathematics. We will get a glimpse of this subject, and the many levels of infinity, and we will see that the infinite is even more magnificent than one might ever have imagined.