Abstract: In recent years there have been good reasons to believe that cosmology with a positive cosmological constrant is the best fit to observable phenomena in astronomy. The cosmological constant as a measure of the energy desity of the vacuum is too small at the particle physics level, but it is comparable to the present matter density. However,the small cosmological constant and cosmic coincidence problem cannot be resolved within the known laws of physics, which is the famous cosmological constant problem in physics and astronomy. In this review we first introduce the great progress in observational evidence for the accelerating universe with the positive cosmological constant and discuss a number of approaches to this puzzle, which at this point remains one of the most significant unsolved problems in fundamental physics.