This paper estimates and solves a multi-country version of the standard DSGE New Keynesian (NK) model. The country-specific models include a Phillips curve determining inflation, an IS curve determining output, a Taylor Rule determining interest rates, and a real effective exchange rate equation. The IS equation includes a real exchange rate variable and a countryspecific foreign output variable to capture direct inter-country linkages. In accord with the theory all variables are measured as deviations from their steady states, which are estimated as long-horizon forecasts from a reduced-form cointegrating global vector autoregression. The resulting rational expectations model is then estimated for 33 countries on data for 1980Q1-2006Q4, by inequality constrained IV, using lagged and contemporaneous foreign variables as instruments, subject to the restrictions implied by the NK theory. The multi-country DSGE NK model is then solved to provide estimates of identified supply, demand and monetary policy shocks. Following the literature, we assume that the within country supply, demand and monetary policy shocks are orthogonal, though shocks of the same type (e.g. supply shocks in different countries) can be correlated. We discuss estimation of impulse response functions and variance decompositions in such large systems, and present estimates allowing for both direct channels of international transmission through regression coefficients and indirect channels through error spillover effects. Bootstrapped error bands are also provided for the cross country responses of a shock to the US monetary policy. JEL Classification: C32, E17, F37, F42.