Teams does not have a simple "export this chat" button sitting in the interface for a work or school account, which surprises a lot of people the first time they go looking for one. What is actually available depends on whether you need this for yourself or need it as an administrator pulling records for compliance or a legal request, and the right method is different for each.
For your own chats, print to PDF
Open teams.microsoft.com in Edge or Chrome, open the chat you want to save, and scroll up to load the full history you need, since Teams only loads recent messages until you scroll. Select all the text with Ctrl+A, then print with Ctrl+P and set the destination to Save as PDF. This captures the visible conversation, including images and shared files as thumbnails, in a format anyone can open later.
For a personal Microsoft account
If you are signed in with a personal account such as Outlook.com or Hotmail rather than a work account, you can request a full export of your data, including Teams chats, through Microsoft's own privacy and data export tools tied to that account.
For administrators pulling records across the organization
This is where Microsoft Purview comes in. Under the Audit section of Purview, an admin can search chat and channel activity by date range, user, and activity type, then export the results as CSV or PST. This is the appropriate path for legal holds, HR investigations, or compliance requests, since it captures message content the way an organization is required to be able to produce it, rather than relying on a manual screenshot style export.
For developers building an integration, Microsoft also publishes dedicated Teams Export APIs that can pull one to one, group, meeting, and channel messages programmatically, which is the right tool if you are exporting at scale on an ongoing basis rather than doing a one time pull.
If your organization needs a proper compliance or eDiscovery process set up around Teams rather than one off exports, our Microsoft 365 and Modern Work team can help you build that. Get in touch if that is what you need.