Even though BitTorrent has pretty much completely dominated file sharing, other P2P networks and protocols are still around. You may have once used GNUtella, or even eDonkey, but Direct Connect and SoulSeek are two less-obvious methods for sharing files.
This page is split into sections for each type of transfer protocol/network. Since there are really no clear industry-leading clients for any of these networks/protocols (other than LimeWire, which is here), I ranked the apps by the number of users and general ease-of-use.
GNUtella was the answer to the FastTrack network, used by the early popular file-sharing clients like Napster and later KaZaa. GNUtella decentralized the location of the files being shared from a single server to all the clients. This let GNUtella continue operations, while the older networks were being shut down for copyright infringement. The most popular GNUtella client, LimeWire, is still heavily used today, even though BitTorrent has since taken the spotlight.
LimeWire (v. 5.2.13)
LimeWire was perhaps the most popular P2P file-sharing app before the BitTorrent explosion. It is similar to KaZaa, much of which was modeled after the original file-sharing rock star, Napster. Like those applications, LimeWire was plagued by malware. If you still use GNUtella for P2P fileshareing, you should really consider giving Linux a spin. That's right, Linux has LimeWire, and no viruses.
FrostWire (v. 4.18.1)
FrostWire is essentially the same thing as LimeWire. They both take after KaZaa and use the GNUtella network.
FrostWire has a flashier tabbed interface, but it's a noticeably bulkier app. Also, the search results are absolutely inundated with porn files. It doesn't matter what you're searching for, these files will dominate your results.
The eDonkey network picked up steam immediately after Napster's destruction and again had a burst of popularity after KaZaa was compromised. However, due to GNUtella clients picking up where the FastTrack apps left off, and having almost identical interfaces, eDonkey's popularity was short lived.
aMule (v. 2.2.4)
aMule has a great interface, and it had large list of servers that were available to connect to right after the installation. Surprisingly, many of these servers were heavily populated by users and files. Unfortunately, my router/firewall prevented me from successfully connecting to any of them.
MLDonkey (v. 2.9.5)
MLDonkey has a fantastic, user-friendly interface. However, it failed to connect to any servers, and, unlike Amule, it didn't even populate a list of available servers. This leaves you to deal with any router/firewall issues before finding out if there are any good servers available.
Direct Connect Network
Direct Connect has been around for almost a decade, but it never really took off. It is a P2P network like GNUtella and eDonkey, but relies on 'hubs' to route the P2P interactions.
The reason Direct Connect never caught on is likely due it's unfamiliar interface and archaic networking settings. Despite being passed over for more user-friendly protocols in it's day, Direct Connect networks now have a healthy number of users.
LinuxDC++ (v. 1.0.2)
Finding and connecting to a Direct Connect hub was not difficult at all with LinuxDC++. The default list of public hubs was full upon installation, and there are peers brimming with available files from all over the globe. The interface is simple and uncluttered, making LinuxDC++ incredibly user-friendly.
Valknut (v. 0.3.13)
Valknut opens a desktop-like outer window, which contains features within their own windows, much like a Java application. The list of public hubs was empty after installation. This app will require you to find hub addresses externally.
SoulSeek is a relative newcomer to file-sharing. It was created in '02/'03, like so many others, with the purpose of picking up where Napster left off. SoulSeek uses centralized servers for handling file searches and communications between users.
Unlike BitTorrent, you must transfer entire files from a single user with SoulSeek. While this clearly limits the number of files and speed of transfers, it fosters the community approach to file-sharing.
Nicotine+ (v. 1.2.9)
When Nicotine+ first starts up, you'll need to input a user name and password in order to connect to the server. After your information has been accepted, you'll be able to enter chat rooms, do private chats, browse other users' shared files, and search for files.
There are actually a surprising number of files available via Nicotine+, but they're mostly music files. SoulSeek offers users the ability to create and join chat rooms whose topics are usually genre-driven, such as “drum'n'bass” and “DEATH METAL CLUB.” This allows users to communicate and share the music being discussed, which could be a good way to find new or unsigned material. Unfortunately, more often than not it becomes another back-and-forth flaming session.