  The main subjects of the paper is studying the fundamental groups of closed
symplectically aspherical manifolds. Motivated by some results of Gompf, we
introduce two classes of fundamental groups $\pi_1(M)$ of symplectically
aspherical manifolds $M$ with $\pi_2(M)=0$ and $\pi_2(M)\neq 0$. Relations
between these classes are discussed. We show that several important classes of
groups can be realized in both classes, while some of groups can be realized in
the first class but not in the second one. Also, we notice that there are some
interesting dimensional phenomena in the realization problem.
  The above results are framed by a general research of symplectically
aspherical manifolds. For example, we find some conditions which imply that the
Gompf sum of symplectically aspherical manifolds is symplectically aspherical,
or that a total space of a bundle is symplectically aspherical, etc.
