Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. This talk will extend these calculations to temporally varying environments (periodic and stochastic). Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. The invasion speed in a variable environment is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. Sensitivity analysis provides expressions that include stochastic, periodic, and time-invariant environments as special cases.