Is there an integrated spell checker?

Yes, Overleaf provides a free multi-language spell check facility for all users.

To change the language used by the spell checker, go to your profile settings menu and select your desired language. In the editor any incorrectly spelled words will be highlighted with a red underline:

You can right click on any highlighted word to display a list of spelling suggestions. Note that the spell check highlights are only displayed in the editor pane (the left hand window), not in the compiled pdf.

(If you're using Vim on Overleaf, you can use the keyboard shortcut z= over a word highlighted as a typo to show spelling suggestions.)

To change the spell check language on an individual document (rather than globally for all projects), simply click the 'gear' icon in the top right of the editor to open the project settings menu, and choose the spell check language from the dropdown:

For a full list of languages that are supported by the integrated spell checker on Overleaf, please see this FAQ answer.