The paper describes the general features and trends of the electrostatic assembly (EA) of block polyelectrolytes. We performed computer simulations of the associative behavior of aqueous mixtures of diblock copolymers containing one neutral water-soluble block and one either positively or negatively charged polyelectolyte (PE) block. While the neutral block is readily soluble in water, the hydrophilic vs hydrophobic nature of the neutral backbone of the PE block and the compatibility of the blocks vary in a broad range. We investigated the role of (i) electrostatics, (ii) solvophobicity of the PE block, (iii) compatibility of the polymer blocks, and also (iv) compatibility of small ions with the polymer blocks. We employed the dissipative particle dynamics (DPD) method and used the generally recognized formula (J. Chem. Phys. 1997, 107, 4423) for recalculating the Flory-Huggins interaction parameters in DPD parameters of soft coarse-grained repulsion forces. The Coulomb interactions are described by the rigorous expression derived for the exponentially smeared charge with a fairly low charge decay length lambda = 0.2. A low lambda value has been used to reproduce the behavior of small counterions as realistically as possible at the DPD level. We compared the self-assembling behavior of charged and neutral copolymers for all the systems. The conclusions of the study can be briefly outlined as follows: Even though long-range electrostatic interactions are a prerequisite for electrostatic self-assembly and the increase in entropy due to liberation of mobile counterions represents an important driving force in all cases, the solvent quality for the PE backbone and incompatibility of blocks play an important role and substantially modify the association process. Only dimers containing one positively and one negatively charged chain are formed in systems with readily soluble PE blocks. The formation of large coreshell associates assumes (in addition to the effect of electrostatics) significantly unfavorable interactions of PE segments with water and with segments of the water-soluble block. The presence of opposite charges on different chains promotes the formation of associates, i.e., both the fraction of associates and association number increase, but the latter increase is fairly small (taking into account the value that is attained in corresponding neutral system).