The Viking Gymir waited for us in Budapest, her sleek lines reflecting the lights of the Chain Bridge as evening settled over the Danube. Over the next two weeks, she would carry us through the heart of Europe—from Hungary's grand capital through Austria's vineyard-lined valleys, past the fairy-tale castles of the Rhine Gorge, and finally to Amsterdam's canal-laced streets. But even as we disembarked in the Netherlands, our journey wasn't complete. Four more days awaited us in Paris, turning what began as a river cruise into an epic traverse of Europe's most storied waterways and cities.