This way is working for me on Tomcat 6 (See jevelopers answer for tomcat 7):
curl --upload-file <path to warfile> "http://<tomcat username>:<tomcat password>@<hostname>:<port>/manager/deploy?path=/<context>&update=true"
Example:
curl --upload-file targetdebug.war "http://tomcat:tomcat@localhost:8088/manager/deploy?path=/debug&update=true"
Very easy peasy. Output is like this:
OK - Undeployed application at context path /debug
OK - Deployed application at context path /debug
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