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We will give a short overview of the recent breakthrough results of Y. Zhang, J. Maynard and T. Tao proving that there are infinitely many bounded gaps between consecutive primes. The results of Maynard and Tao even imply that there are arbitrarily long finite blocks of consedcutive primes in bounded intervals infinitely often. We will mention several other results which refer to so called Polignac numbers which have the property that they appear infinitely often as difference between consecutive primes. We also mention answers on several 60-70 years old conjectures of Erdös referring for the difference of consecutive primes. Finally we mention that the celebrated theorems of Green-Tao and Zhang-Maynard-Tao can be generalized in the way that there exists a bounded difference d with the property that there are arbitrarily long finite arithmetic progressions of primes such that p+d is the prime following p for every element of the progression.