How incomputable is Kolmogorov complexity?

Abstract: Kolmogorov complexity is the length of the ultimately compressed version of a file (that is, anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. We discuss the incomputabilty of Kolmogorov complexity, which formal loopholes this leaves us, recent approaches to compute or approximate Kolmogorov complexity, which approaches are problematic and which approaches are viable.