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Design methods are, for the first time, suggested for the complex impedance transformers, to solve the conventional problem of imaginary value of characteristic impedance of transmission-line sections. The complex impedance transformers consist of only transmission-line sections, and the design formulas are the function of the reίection coefficients of the complex termination impedances. Firstly, the impedance transformers that transform a complex impedance into a real one are discussed. Secondly, those transforming a complex impedance to another complex one are studied. For the impedance transformers of the first type, the regions of the Smith chart where one complex impedance can be transformed into one real one by only one transmission-line section are called allowed regions, while the outside forbidden regions. Depending on where the complex impedances are located, a method to reduce the size is discussed in more detail. For the second case where both termination impedances are complex, the complex conjugate of one of two complex impedances should be located on a Smith chart for maximum power transfer. The design formulas are then obtained in a similar manner. To validate the design formulas, three impedance transformers terminated in a fixed complex impedance of (1000 - j30) Ω and three real impedances of 50, 60, and 80 Ω are fabricated and measured. The measured results show quite good agreement with prediction, return losses of 30-38 dB being achieved around the design center frequency of 2 GHz.