Speaking Thursday in Berlin, German Chancellor Angela Merkel said refugees would be welcomed -- but would need to assimilate into society, respect the government and the values of the country, and learn German.
And following a meeting in Brussels, Belgium, the European Council -- the heads of government of the 28 European Union countries -- said the biggest test still lies ahead.
"It is clear that the greatest tide of refugees and migrants is yet to come," said Donald Tusk, president of the European Council.
Tusk said that European leaders had agreed to provide more help to Lebanon, Jordan, Turkey and other countries in the region in the hopes of at least inducing some people who are fleeing conflicts and poverty to stay in the Middle East.
Tusk said 1 billion euros (about $1.12 billion) would be committed to the region in cooperation with the U.N. High Commissioner for Refugees and the World Food Program.
Many among the wave of people landing on European shores each day are from Syria, where a 4½-year civil war has reduced cities to rubble and wiped out any means of earning a living.
In addition to supplying more aid to Middle Eastern countries, Tusk said the national leaders meeting in Brussels had agreed to strengthen the protection of the EU's outer borders.
The leaders had agreed, Tusk said, that "the current chaos at our external borders must end."
"Proper management and control of our external borders is our common responsibility," he said. "It would be unfair to put all the burden on Italy or Greece or other countries."
Still, the EU is a long way from agreeing on a united and coordinated response.
"The measures we have agreed today will not end the crisis," Tusk said. "But they are all necessary steps in the right direction."
For all that show of commitment, the final statement by the national leaders made no mention of a tentative agreement reached earlier by lower-level officials -- albeit with fierce dissent -- for countries to accept 120,000 migrants across the EU
Tusk said European leaders would meet again in October.