A long-standing problem in international macroeconomics has been to understand exchange rates. There are two broad aspects to the macroeconomics of exchange rates. The first is to understand the determinants of exchange rates and the other is to understand how exchange rates impact the macro economy. This dissertation focuses on improving the profession's understanding of exchange rates by studying both of these relationships.  Chapter 1 studies how a country's choice of monetary policy and its associated exchange rate regime impacts a country's welfare. The analysis is conducted using a three-country dynamic stochastic general equilibrium (DSGE) exchange rate model with nominal price stickiness á la Calvo (1983). Under complete markets, the managed float dominates under local-currency pricing (LCP), but the fix dominates under producer-currency pricing (PCP). Under incomplete markets, the choice of exchange rate regime is nearly irrelevant for welfare whether export prices are set by LCP or PCP. The presence of third-country shocks are also largely irrelevant to the choice of exchange rate regime.  Chapter 2 studies a country's choice to remain in or to exit a currency union, such as the European Monetary Union. Welfare rankings for a single country exit, complete dissolution, and preservation of the union are examined under a variety of economic environments. Using a three-country DSGE model and simple Taylor-type monetary policy rules, remaining in the currency union is consistently the least best option. Unless physical capital is included in the model, however, the magnitude of the welfare differences across alternative policy regimes is very small. Although the behavior of macroeconomic aggregates differs across regimes, the monetary policy choices considered may be largely irrelevant to welfare.  Chapter 3 addresses the exchange-rate disconnect puzzle (joint work with Nelson Mark). Predictive regressions for bilateral exchange rates are typically run using variables from the associated bilateral country pairs. These regressions characteristically show limited explanatory power. Considerable relative increases in adjusted R2 are obtained by augmenting the regressions with third-country variables. A three-country exchange rate model is presented in which cross-country heterogeneity opens up channels for third-country effects to influence the bilateral rate.