It is very difficult to pinpoint where the Warsaw city center is. Varsovians differ in its definition. Some would say it is the area around the Palace of Culture and Science (and the Central Railway Station), which bustles with fashionable shops and hotels. Others would argue that plac Trzech Krzyży is the center—or plac Zbawiciela. For many, it would be the area around the Royal Castle and the Old Town square; after all, it was at this location that a fishing hamlet was founded and eventually grew into Warsaw town.
For the visitor, the question is, happily, not so important because all these places are not at all far from one another, and Warsaw is an easy city to navigate. The Palace of Culture and Science will certainly provide you with a useful orientation point: to its north lies the Old Town, which encompasses most of the Royal Route; to its south, the Diplomatic Quarter and the Łazienki Park. West of the Old Town lie Muranów, Mirów, and Wola, neighborhoods in the former Jewish district. All these sights are on the left bank of the Vistula River.
On the right bank is the Praga District, a poorer quarter of workers and artisans that emerged from the war fairly intact. Today, Praga is becoming increasingly fashionable, and many visitors find its galleries, bars, and unique "provincial" flavor well worth the trip across the Vistula.