We describe measurements of single–cycle terahertz pulse propagation in a random medium. The unique capabilities of terahertz time–domain spectroscopy permit the characterization of a multiply scattered field with unprecedented spatial and temporal resolution. With these results, we can develop a framework for understanding the statistics of broadband laser speckle. Also, the ability to extract information on the phase of the field opens up new possibilities for characterizing multiply scattered waves. We illustrate this with a simple example, which involves computing a time–windowed temporal correlation between fields measured at different spatial locations. This enables the identification of individual scattering events, and could lead to a new method for imaging in random media.