  The halting probability of a Turing machine,also known as Chaitin's Omega, is
an algorithmically random number with many interesting properties. Since
Chaitin's seminal work, many popular expositions have appeared, mainly focusing
on the metamathematical or philosophical significance of Omega (or debating
against it). At the same time, a rich mathematical theory exploring the
properties of Chaitin's Omega has been brewing in various technical papers,
which quietly reveals the significance of this number to many aspects of
contemporary algorithmic information theory. The purpose of this survey is to
expose these developments and tell a story about Omega, which outlines its
multifaceted mathematical properties and roles in algorithmic randomness.
