Techniques for reducing power consumption in digital circuits have become increasingly important because of the growing demand for portable multimedia devices. Digital filters, being ubiquitous in such devices, are a prime candidate for low-power design. We present a new algorithmic approach to low-power frequency-selective digital filtering which is based on the concepts of adaptive approximate processing. This approach is formalized by introducing the class of approximate filtering algorithms in which the order of a digital filter is dynamically varied to provide time-varying stopband attenuation in proportion to the time-varying signal-to-noise ratio (SNR) of the input signal, while maintaining a fixed SNR at the filter output. Since power consumption in digital filter implementations is proportional to the order of the filter, dynamically varying the filter order is a strategy which may be used to conserve power. From this practical technique we abstract a theoretical problem which involves the determination of an optimal filter order based on observations of the input data and a set of concrete assumptions on the statistics of the input signal. Two solutions to this theoretical problem are presented, and the key results are used to interpret the solution to the practical low-power filtering problem. We construct a framework to explore the statistical properties of approximate filtering algorithms and show that under certain assumptions, the performance of approximate filtering algorithms is asymptotically optimal.