Danzig was a German city in what is now part of Poland. The Polish city of Gdansk is to Danzig what Istanbul is to Constantinople. The question is, is the city legitimately German or Polish?
The Poles claim that they originally settled the area, and that the first references to Gdansk emerged around 997, when a missionary, Bishop Adalbert of Prague, traveled to the "urbs Gyddanyzc" and was killed by the pagan Prussians. The Germans, however, counter that Scandinavians and Gothic tribes first settled the area.
In the 1100s and 1200s the Pomeranian nobles encouraged German settlers to come east and settle the empty spaces in their realm. In 1224 Danzig was given city status. The Polish Duke of Masovia called upon the Teutonic Knights of the Cross to fight the pagan Prussians to the north. The knights were free to Christianize and colonize any lands not under Polish control. Under the Treaty of Solin, Danzig came under the rule of the German Order of Knights on September 13, 1309. Then in the Battle of Tannenburg in 1410, the Teutonic knights lost Danzig to a force of Poles and Lithuanians. This is where things get complicated. Danzig entered a security alliance with the crown of Poland, but became a member of the German Hanseatic League. Danzig was essentially a free city-state. Now, we can't just go back in time and ask the people there whether they were German or Polish. The truth is probably that there was always a mixture of German and Slavic peoples living in the city.
In 1793, Danzig lost its independence when Prussia, Russia, and Austria carved up Poland. Danzig reluctantly joined Prussia on May 5, 1793. In 1807 the Prussians surrendered to Napoleon Bonaparte, and Danzig nominally regained its status as a free city, however, it was occupied by French troops, who ransacked the city. Following Napoleon's defeat, in 1814 Danzig reverted to Prussian control.
In 1871, Prussia became the German Empire. This empire, the Second Reich, was doomed by its entry into the Triple Alliance. After every German strategist's worst nightmare, a two-front war of attrition, Germany capitulated in 1918. With the signing of the Treaty of Versailles, Danzig came under Allied control. The Free City of Danzig was re-established on November 15, 1920. It was not under Polish control, yet it was meant to give Poland access to the sea. This "Polish corridor" separated East Prussia from the rest of Germany, which infuriated Hitler. Hitler invaded Poland, and Danzig with it, in 1939, starting World War II.
Around 1942-43 everything started going horribly wrong for Hitler. After their victory at Stalingrad, the Soviets began advancing rapidly through German-occupied territory, eventually reaching East Prussia, part of Germany itself. Danzig was "liberated" by the Soviets in March 1945. Now, the Soviets were pissed at the Germans. Mighty pissed. Something like 20 million Soviets were killed in World War II. So as they advanced through Eastern Europe, the Russians basically took out their anger on everyone and everything German. Germans, especially in East Prussia and Danzig, lost their land as it was ethnically cleansed. Hundreds of thousands of Germans were killed, raped, and deported. Danzig was torched and lost at least 25% of its population.
To appease Stalin, Roosevelt and Churchill ignored the atrocities the Soviets were committing. They also ignored part of the Treaty of Versailles, which established Danzig as an independent city. At the Yalta conference it was decided that Danzig would become part of Poland. It is at this point that the history of Danzig ends and the history of Gdansk begins.
So according to the Government-in-Exile of the Free State of Danzig, the city should be re-established in accordance with international law. After all, it does technically still exist. But for better or for worse, the German population has been driven out for almost 60 years now. Gdansk is now unquestionably Polish, and the Poles who live there cannot be forced out. Three wrongs don't make a right. However, I think the Polish government at least owes an apology to the ethnic Germans who were forced out after World War II.
For two different views, see:
The official site of Gdansk
Free State of Danzig Government-in-Exile