  Puzzles often give birth to the great discoveries, the false discoveries
sometimes stimulate the exiting ideas in theoretical physics. The historical
examples of both are described in Introduction and in section ``Cosmological
Puzzles''. From existing puzzles most attention is given to Ultra High Energy
Cosmic Ray (UHECR) puzzle and to cosmological constant problem. The 40-years
old UHECR problem consisted in absence of the sharp steepening in spectrum of
extragalactic cosmic rays caused by interaction with CMB radiation. This
steepening is known as Greisen-Zatsepin-Kuzmin (GZK) cutoff. It is demonstrated
here that the features of interaction of cosmic ray protons with CMB are seen
now in the spectrum in the form of the dip and beginning of the GZK cutoff. The
most serious cosmological problem is caused by large vacuum energy of the known
elementary-particle fields which exceeds at least by 45 orders of magnitude the
cosmological vacuum energy. The various ideas put forward to solve this problem
during last 40 years, have weaknesses and cannot be accepted as the final
solution of this puzzle. The anthropic approach is discussed.
