By default, SharePoint 2010 will not let you show html pages in the browser.
If you add them to your site, either using upload or SharePoint Designer, the browsers will prompt you to save the file.
It is easily remedied; open central admin, manage web applications, select web application, general settings, and change "Browser file handling" to permissive.
Ah, but what if you do not have access to CA - like in Office 365?
The official answer is simple: "Not supported"
If you can rename the files, you may still be in luck. Rename test.htm(l) to test.aspx.
Simple example: http://husmo.sharepoint.com/klokke.aspx (best viewed with an html 5 browser)