Google Dataset Search does just what the name says. Why not start there?
Kaggle is an alternative.
DataHub.io also allows thematic search of datasets.
Wikipedia keeps a list of data portals established by governments and international organisations.
The UN Statistics office references their various databases.
Open research data repositories contain many research datasets. Here is a list by subject.
IPCSR (Inter-university Consortium for Political and Social Research) is a major social science data host.
It also hosts a social media archive.
The Qualitative Data Repository is self-explanatory.
Data is Plural is a newsletter with interesting data sources.
NASA's Earth Observation Data: https://www.earthdata.nasa.gov/
GitHub's list of Awesome Public Datasets.
Project Implicit's data on implicit social cognition (bias based on race, age, ability, gender, etc.)
Wikimedia and its contributors are building Wikidata.