Dynamic systems of an arbitrary real order (fractional-order systems) are considered, A concept of a fractional-order PIlambda D-mu-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffer-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller both for the open and closed leap. An example demonstrating the use of the obtained formulas and the advantages of the proposed PIlambda D-mu-controllers is given.