A big challenge in symplectic geometry is the search for irreducible symplectic manifolds. After O'Grady constructed striking examples out of singular moduli spaces of sheaves on projective abelian and K3 surfaces for special nonprimitive Mukai vectors v with v.v=8, Kaledin, Lehn and Sorger proved that for all nonprimitive Mukai vectors v with v.v>8 the moduli space is not symplectically resolvable if the ample divisor is general. In this thesis we investigate the remaining cases of moduli spaces of semistable sheaves on projective K3 surfaces - the cases of Mukai vector (0, c,0) as well as...