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.
Datacite Commons may also be of interest.
RE3data is a repository of data repositories. You can use it to find a repositories hosting data on the subject you are interested here.
Wikipedia keeps a list of data portals established by governments and international organisations.
NB: US Government data deleted or threatened by the Trump government may be found in LUMOS, an emergency repository managed by ICPSR.
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.