Wireless systems often make use of the quadrature relationship between pairs of signals to effectively cancel out-of-band and interfering in-band signal components. The understanding of these systems is often simplified by considering both the signals and system transfer functions as "complex" quantities. The complex approach is especially useful in highly integrated multistandard receivers where the use of narrow-band fixed-coefficient filters at the RF and high IF must be minimized. This paper first presents a tutorial review of complex signal processing for wireless applications. The review emphasizes a graphical and pictorial description rather than an equation-based approach. Next, a number of classical modulation architectures are described using this formulation. Finally, more recent developments such as complex filters, image-reject mixers, low-IF receivers, and oversampling analog-digital converters are discussed.