We advance a coherent development of zero-crossing-based methods and theory appropriate for fast signal analysis. Quite a few ideas pertaining to zero-crossing counts found in the literature can be expressed and interpreted with the help of this more general setup. A central issue addressed in some detail is the fruitful connection which exists between zero-crossing counts and linear filtering. This connection is explored and interpreted with the help of a certain zero-crossing spectral representation, and is then applied in spectral analysis, detection, and discrimination. Zero-crossing counts in filtered time series are called higher order crossings. The theme of this work is that higher order crossings analysis provides a useful descriptive as well as an analytical tool that can in many respects match spectral analysis. To a great extent these two types of analysis are, in fact, equivalent, but each emphasizes a different point of view. Advantages offered by higher order crossings are great simplicity and a drastic data reduction.