Originally Posted by
buzzingrobot
The overwhelming majority of Ubuntu packages are migrated over from Debian. Even more of Mint is built directly on Ubuntu packages. Applications available in one of those distributions are almost always available for installation in the usual way in the others, although the specific versions can vary from release to release.
Installing a package is seldom a matter of just copying a single file. Applications usually contain any number of files, and the packages that we use to install them include data and scripts to make sure all the pieces end up in the right spots in the filesystem. Any other packages -- the dependencies -- needed will be downloaded and installed *if we use* programs created to do that, like the Software Center, apt, Synaptic, etc.
Debian packages can be searched at packages.debian.org. Ubuntu packages can be searched at packages.ubuntu.com. Both show a package's dependencies, changelog, etc.
Mint packages are at packages.linuxmint.com. Since Mint relies on Ubuntu repos, there are relatively few packages there, so they're browsable instead of searchable.
[EDIT: But, yes, technically, packages can be downloaded and installed locally using Software Center, apt, etc. That should work using an Ubuntu package on Ubuntu, a Mint package on Mint, etc. Trying to install a foreign package is risky for a number of reasons.]
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