Kolmogorov complexity of a of a binary word, as Kolmogorov himself has defined it, is the length of the shortest program producing that word. Kolmogorov tried and succeeded to use that definition to create a new informatiom theory in parallel to Shannon's one. But he also intended to define on that base the notion of randomness. Unfortunately, it turned out that the original definition is not good for that goal (in particular, to define what a random infinite sequence is). To get a correct definition of an infinite binary sequence one needs to consider Kolmogorov complexity in a more broad sense: namely, as the length of the shortest description under some appropriate description mode.