Abstract El Niño–Southern Oscillation (ENSO) in a century-long integration of a Bureau of Meteorology Research Centre (BMRC) coupled general circulation model (CGCM) drives rainfall and temperature changes over Australia that are generally consistent with documented observational changes: dry/hot conditions occur more frequently during El Niño years and wet/mild conditions occur more frequently during La Niña years. The relationship between ENSO [as measured by Niño-4 or the Southern Oscillation index (SOI), say] and all-Australia rainfall and temperature is found to be nonlinear in the observations and in the CGCM during June–December: a large La Niña sea surface temperature (SST) anomaly is closely linked to a large Australian response (i.e., Australia usually becomes much wetter), whereas the magnitude of an El Niño SST anomaly is a poorer guide to how dry Australia will actually become. Australia tends to dry out during El Niño events, but the degree of drying is not as tightly linked to the magnitude of the El Niño SST anomaly. Nonlinear or asymmetric teleconnections are also evident in the western United States/northern Mexico. The implications of asymmetric teleconnections for prediction services are discussed. The relationship between ENSO and Australian climate in both the model and the observations is strong in some decades, but weak in others. A series of decadal-long perturbation experiments are used to show that if these interdecadal changes are predictable, then the level of predictability is low. The model’s Interdecadal Pacific Oscillation (IPO), which represents interdecadal ENSO-like SST variability, is statistically linked to interdecadal changes in ENSO’s impact on Australia during June–December when ENSO’s impact on Australia is generally greatest. A simple stochastic model that incorporates the nonlinearity above is used to show that the IPO [or the closely related Pacific Decadal Oscillation (PDO)] can appear to modulate ENSO teleconnections even if the IPO–PDO largely reflect unpredictable random changes in, for example, the relative frequency of El Niño and La Niña events in a given interdecadal period. Note, however, that predictability in ENSO-related variability on decadal time scales might be either underestimated by the CGCM, or be too small to be detected by the modest number of perturbation experiments conducted. If there is a small amount of predictability in ENSO indices on decadal time scales, and there may be, then the nonlinearity described above provides a mechanism via which ENSO teleconnections could be modulated on decadal time scales in a partially predictable fashion.