David Hilbert, a 20th century logician, famously said: “The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification….” As pointed out by Hilbert, the concept of infinity has been an important topic in philosophy since the time of early Greek philosophers. In addition to its philosophical importance, it also has significant mathematical aspects. For example, the concept forms the basis of some counter-intuitive mathematical results, such as being able to paint an infinite surface with a finite amount of paint. Thus, a thorough investigation of the concept of infinity will need to have both a philosophical dimension and a mathematical/logical dimension. This proposed course aims to provide such an investigation. The investigation will be carried out in three overlapping dimensions: historical, metaphysical, and mathematical. In the first dimension, the development of the concept of infinity will be traversed, starting from early Greek thought and ending with 20th century views on infinity. The main focus of the metaphysical dimension of the course will be the distinction between potential infinity and actual infinity. The arguments for and against the existence of actual infinity will be thoroughly examined. In addition to these arguments, the metaphysical dimension of the course will also include an analysis of human finitude and the possibility of infinite beings. The mathematical dimension will cover early paradoxes of motion and time, Galileo’s “paradox”, Torricelli’s trumpet, Cantor’s transfinite mathematics, and some theorems relevant to the discussion of infinity.