1. Go to https://developers.google.com and login or create an account, if necessary.
2. Go to https://console.developers.google.com/project and click on Create Project: 
3. You’ll be redirected to this screen where you need to fill in the project information and click Create: 
4. After creating your project, select it and click on Google APIs link to go to API library: 
5. Find YouTube Data API in the API library and select it:
6. Click Enable:
7. Click Create Credentials:
8. Fill in the fields and click What Credentials do I Need:

You’ve got your YouTube API key!

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