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A systematic description of recent measurements of the real Rs and imaginary Xs parts of the microwave surface impedance Rs+ iXs in high-quality high-Tc, superconducting single crystals, namely YBa2Cu3O6.95, Ba0.6K0.4BiO3, TI2Ba2CaCu2O8−δ, Tl2Ba2CuO6+δ, and Bi2Sr2CaCu2O8, is given. Electrodynamic principles of experimental techniques and uncontrollable factors that affect the accuracy of Rs(T) and Xs(T) measurements are analyzed. Common and distinctive features in the temperature dependences of the surface impedance and complex conductivity in various high-Tc crystals are discussed. The experimental data are interpreted in terms of the two-fluid model taking into account scattering of quasiparticles and characteristic changes in the density of superconducting carriers as a function of temperature. The existing microscopic models of the microwave response of high-Tc superconductors are considered and prospects for further research are outlined.