No other city in the world has as high a concentration of Art Nouveau buildings as Riga. Between 1899 and 1914, roughly a third of the city's entire building stock was put up in the Jugendstil style — some 800 buildings survive today, far more than in Paris or Vienna.
The densest cluster is on Alberta iela and Elizabetes iela, a short walk north of the Old Town. Alberta iela is a showpiece — virtually every building on the street is an Art Nouveau masterpiece, many designed by Mikhail Eisenstein (father of the film director). The facades are extraordinary: writhing human figures, screaming masks, elaborate floral ornament, and allegorical reliefs stacked up five storeys. Walk the whole street slowly and look above the ground floor.
Elizabetes iela has more variety — some National Romanticism (the Latvian variant, which draws on folk motifs rather than Viennese ornament) alongside pure Jugendstil. The Riga Art Nouveau Museum at Alberta 12 occupies a preserved apartment that gives a sense of how these buildings were lived in.
The tourist office on Rātslaukums sells an Art Nouveau walking map; it's a worthwhile investment if you plan to explore seriously.