The idea of associating a free resolution to a finitely generated module was introduced by Hilbert. In the local and graded cases, the module has a minimal free resolution, which is unique up to an isomorphism. Hilbert proved Hilbert’s Syzygy Theorem, which says that the minimal free resolution of every finitely generated graded module over a polynomial ring is finite. Since then, there has been a lot of progress on the structure and properties of finite free resolutions. Much less is known about the properties of infinite free resolutions. Such resolutions occur abundantly since most minimal free resolutions over a graded non-linear quotient ring of a polynomial ring are infinite. The talk will survey results and open problems on infinite free resolutions.