  This is mainly a review of my results related to the title. We discuss, how
many elliptic fibrations and elliptic fibrations with infinite automorphism
group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically
closed field can have.
  This was the subject of my talk at Oberwolfach Workshop "Higher dimensional
elliptic fibrations" in October 2010.
