Installing Red5 by using the .war file is easy on all OS‘es once you have Java and Tomcat running. On a Windows system it is especially easy because the Java and Tomcat Setups create all necessary environment variables and registry entries. However, there are two little problems:
You better not use the default “C:\Program Files\...” installation paths. Apart from problems with spaces in pathnames, you probably don’t want your websites located somewhere deep in “C:\Program Files\ ...” In order to change the default installation path, you actually must read the messages in the Setup dialogs before clicking “Next”. Now that is what I would call a tough job

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Even worse. If you want to upgrade from an older version of Java and/or Tomcat, you are in trouble. Best approach is to backup the webapps folder in Tomcat, and remove Tomcat and Java (in that order). Then check the environment variables and remove everything that is left pointing to Java or Tomcat, and then reinstall the newer versions of Java and Tomcat (in that order, see below).
But life is simple if you have a fresh system. Follow the instructions below.