Until recently, the percolation phase transitions were
believed to be continuous, however, in 2009, a remarkably
different, discontinuous phase transition was reported in a
new so-called "explosive percolation" problem. Each new link
in this problem is established by a specific optimization
process. We develop the exact theory of this phenomenon and
explain its nature. Applying strict analytical arguments to
a wide representative class of models for the infinite
system size limit, we show that the "explosive percolation"
transition is actually continuous though with an uniquely
small critical exponent of the percolation cluster size.
These transitions provide a new class of critical phenomena
in irreversible systems and processes. We obtain a complete
description of the scaling properties of these second order
transitions. For all these models, we find the scaling
functions and the full set of critical exponents, and, also,
the upper critical dimensions which turn out to be
remarkably low, close to 2.