How to "hibernate a process" under Linux/Windows?

Just like e.g. people can press Ctrl+P when running a VirtualBox guest, it can pause the guest, after pressing again Ctrl+P it will continue right there where it was stopped. Are there any command line commands on Windows/Linux where I can do the same with a process? (the process is e.g.: Firefox, etc.)
Thank you!

For Windows, you want PsSuspend. The utility demands a PID, which is a column you can add in Task Manager.

For Linux, seconding the above. kill -STOP and kill -CONT.

pstop will generally work on Linux and other Unixes which implement /proc-based process control (many System Vs, almost no BSDs); kill -TSTP (-TSTP should be used with interactive processes; -STOP may be better for daemons) will work on most modern Unixes. The reverse operation is prun or kill -CONT, respectively. Note that these are not necessarily equivalent; on Solaris they are entirely distinct operations.