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Can you recommend a replacement for Jira that can be self-hosted? Many companies cannot store their data on a third-party server.

Take a look at Gemini[0] for self-hosted issue management and help desk ticketing.

There's at least Redmine. Uses Textile by default, but there were plugins for Markdown.

Redmine nowadays also comes by default with Markdown.

Menu Administration -> Settings -> General tab

There's a drop down for Text formatting which you can set to Markdown.