The paper is devoted to study the solvability and unique solvability of the Cauchy type problem for two-dimensional systems of functional differential equations. The first part of the paper deals with linear differential systems and results obtained there are used afterwards to investigate the nonlinear differential systems. The author uses a method of differential inequalities and a technique of a priori estimates. All the results obtained for differential systems with general operators on the right-hand side are reformulated and make more precise for those with deviating arguments. The author also introduce several counter-examples at the end of the paper to ensure that the results obtained are unimprovable in a certain sense.