A salient characteristic of river discharge is its temporal variability. The time series of flow at a point on a river can be viewed as the superposition of a smooth seasonal cycle and an irregular, random variation. Viewing the random component in the spectral domain facilitates both its characterization and an interpretation of its major physical controls from a global perspective. The power spectral density functions of monthly flow anomalies of many large rivers worldwide are typified by a “red noise” process: the density is higher at low frequencies (e.g., <1 y⁻¹) than at high frequencies, indicating disproportionate (relative to uncorrelated “white noise”) contribution of low frequencies to variability of monthly flow. For many high‐latitude and arid‐region rivers, however, the power is relatively evenly distributed across the frequency spectrum. The power spectrum of monthly flow can be interpreted as the product of the power spectrum of monthly basin total precipitation (which is typically white or slightly red) and several filters that have physical significance. The filters are associated with (1) the conversion of total precipitation (sum of rainfall and snowfall) to effective rainfall (liquid flux to the ground surface from above), (2) the conversion of effective rainfall to soil water excess (runoff), and (3) the conversion of soil water excess to river discharge. Inferences about the roles of each filter can be made through an analysis of observations, complemented by information from a global model of the ocean‐atmosphere‐land system. The first filter causes a snowmelt‐related amplification of high‐frequency variability in those basins that receive substantial snowfall. The second filter causes a relatively constant reduction in variability across all frequencies and can be predicted well by means of a semiempirical water balance relation. The third filter, associated with groundwater and surface water storage in the river basin, causes a strong reduction in high‐frequency variability of many basins. The strength of this reduction can be quantified by an average residence time of water in storage, which is typically on the order of 20–50 days. The residence time is demonstrably influenced by freezing conditions in the basin, fractional cover of the basin by lakes, and runoff ratio (ratio of mean runoff to mean precipitation). Large lake areas enhance storage and can greatly increase total residence times (100 to several hundred days). Freezing conditions appear to cause bypassing of subsurface storage, thus reducing residence times (0–30 days). Small runoff ratios tend to be associated with arid regions, where the water table is deep, and consequently, most of the runoff is produced by processes that bypass the saturated zone, leading to relatively small residence times for such basins (0–40 days).