  During last two decades it has been discovered that the statistical
properties of a number of microscopically rather different random systems at
the macroscopic level are described by {\it the same} universal probability
distribution function which is called the Tracy-Widom (TW) distribution. Among
these systems we find both purely methematical problems, such as the longest
increasing subsequences in random permutations, and quite physical ones, such
as directed polymers in random media or polynuclear crystal growth. In the
extensive Introduction we discuss in simple terms these various random systems
and explain what the universal TW function is. Next, concentrating on the
example of one-dimensional directed polymers in random potential we give the
main lines of the formal proof that fluctuations of their free energy are
described the universal TW distribution. The second part of the review consist
of detailed appendices which provide necessary self-contained mathematical
background for the first part.
