Is CiviCRM compatible with the CMS system Umbraco?
Umbraco won't work directly with civicrm (they are written in different programming languages, ASP vs. PHP)

What you can do easily is to host your civicrm with one of the supported CMS (drupal, wordpress or joomla) into a subdomain (eg. members.example.org) and switch the main site to umbraco (www.example.org)

the civi site will not use any CMS feature beside handing the login/password, but might want to customise a bit the layout so it looks more or less like your main website (eg set the colors, put your logo), so you can use some civi pages (like donation, membership registration, event registration...) into that sub site without confusing too much your visitors, and add a link to the civi subsite when needed in your main site.

It's not as transparent and nice as having everything on the same site, but plenty of organisations do have different sites for the CRM and the main site, and it seems to work for them