Issue
I would like to run a .NET/.NET Core app on Heroku.
Resolution
.NET and .NET Core are not officially supported languages on Heroku at present. You can find a list of our officially support languages here: https://devcenter.heroku.com/articles/buildpacks#officially-supported-buildpacks
Third-party buildpacks are one option for extending the platform https://devcenter.heroku.com/articles/third-party-buildpacks many of which you can browse at https://elements.heroku.com/buildpacks.
For eg: Searching for .NET in buildpacks marketplace yields the all available .NET related buildpacks. By setting the buildpack to the application, can install extra packages and dependencies at deploy time. Please be aware that third-party buildpacks are an option but the buildpacks themselves are unsupported. They might not be maintained by the original authors and have stopped working. If that's the case, you'll need to contact the original author to see if they can help.
Another option is to try our Container support which may allow more flexibility in setting up dependencies using a Dockerfile https://devcenter.heroku.com/articles/container-registry-and-runtime
Regarding any future support we recommend subscribing to the Heroku Changelog. This will let you know about updates, improvements and deprecations ahead of time so that you can make any necessary adjustments.