How to delete a remote tag?

You can push an 'empty' reference to the remote tag name:

git push origin :tagname

Or, more expressively, use the --delete option (or -d if your git version is older than 1.8.0):

git push --delete origin tagname

Note that git has tag namespace and branch namespace so you may use the same name for a branch and for a tag. If you want to make sure that you cannot accidentally remove the branch instead of the tag, you can specify full ref which will never delete a branch:

git push origin :refs/tags/tagname

If you also need to delete the local tag, use:

git tag --delete tagname

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Background

Pushing a branch, tag, or other ref to a remote repository involves specifying "which repo, what source, what destination?"

git push remote-repo source-ref:destination-ref

A real world example where you push your master branch to the origin's master branch is:

git push origin refs/heads/master:refs/heads/master

Which because of default paths, can be shortened to:

git push origin master:master

Tags work the same way:

git push origin refs/tags/release-1.0:refs/tags/release-1.0

Which can also be shortened to:

git push origin release-1.0:release-1.0

By omitting the source ref (the part before the colon), you push 'nothing' to the destination, deleting the ref on the remote end.

A more straightforward way is

git push --delete origin YOUR_TAG_NAME

IMO prefixing colon syntax is a little bit odd in this situation