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The propagation of Rossby waves with positive and negative frequency, corresponding to eastward and westward phase speeds respectively, is investigated. The techniques used are theoretical analysis, ray tracing, and initial value problems in barotropic and baroclinic numerical models. It is found that the characteristics of positive and negative frequency Rossby waves can differ significantly from each other andfrom those of stationary, zero frequency Rossby waves. However, general deductions from studies of stationary Rossby waves are still found to be valid. Using an analytic Gill-type model and a dry primitive equation model with only idealised vorticity or thermal forcing, a possible trigger mechanism for the Madden Julian Oscillation (MJO) has been studied. The results show that eastward moving forcing in the subtropics or extratropics can lead to a significant equatorial Kelvin wave response which tends to be a maximum in the African/Indian Ocean sector, and is enhanced by easterly winds in the upper troposphere. It is suggested . that one mechanism for initiating the MJO is for eastward moving extratropical waves to excite a large equatorial response, sufficient to trigger large-scale convection, in the presence of favourable easterly winds in the upper troposphere. The dry primitive equation model is used to study the possible interaction of atmospheric flow in the two hemispheres and the triggering of other equatorial waves. It is found that stationary and westward moving forcing in the Northern Hemisphere extratropics can give a significant Southern Hemisphere response. A westward moving forcing in the subtropics, with a period of several days, can trigger the equatorial mixed Rossby-gravity and n=l Rossby waves. The zonal basic flow is found to have a significant effect on these equatorial wave responses.