Council of European Social Science Data Archives (CESSDA)
CESSDA is a consortium of national data archives from various European countries. CESSDA's data catalogue offers a central point of access to the holdings of the individual archives.
Harvard's Dataverse Network is a repository for research data and data collections.
Leibniz Institute for the Social Sciences (GESIS)
GESIS is one the premier data archives in Western Europe. It is the principal repository for collections such as the Eurobarometer series of polls in European Union members and the International Social Survey Programme data on social attitudes and issues. It also has a variety of historic data sources, such as a large collection of data from the German Democratic Republic and a similar collection of studies for pre-WWII Germany. GESIS is also a member of the Council of European Social Sciences Data Archives.
National Bureau of Economic Research (NBER)
In addition to its work on business cycles, the NBER has a data archive that covers topics such as cross-national technology adoption, manufacturing productivity, financial openness and exchange rate regimes, and economic policy uncertainty. Its data archive also covers topics such as labor markets, health economics, and population demographics.
The UK Data Archive is another of the premier data archives in Western Europe, with a wide-ranging collection covering topics such as the finances and trade of the East India Company, the fiscal straits of European governments in the Middle Ages, Orange Order membership in Northern Ireland, and interviews with black immigrants to Britain in the 1800's and 1900's. The UKDA is also a member of the Council of European Social Sciences Data Archives. You can search its holdings at https://discover.ukdataservice.ac.uk/.
As part of its Open Data Initiative, the World Bank has opened access to its data collections with compilation into a single data catalog. The holdings range from larger databases covering multiple topics to more narrowly focused collections associated with particular research projects. You can also search for World Bank data by country or by looking for specific indicators.