Tutorials/Slime farming

Slimes only appear in 10% of chunks from y=39 and below, as determined by your world seed. Various utilities exist to help locate these chunks, for example Amidst 3.6 or later, or http://chunkbase.com/apps/slime-finder. They can also spawn in higher y values in swamps, but building farms in swamps is impractical.

Once a slime chunk has been located, the best farm, and quite simple, is to dig out the chunk, all the way down to bedrock if so desired, and from y=38 start building platforms that cover the entire chunk every 3 blocks down. Eg, y=38, y=35 and so on. Around the chunk platform, make a 3 wide dead fall, and on the lowest level, use outward flowing water streams under the deadfall to direct the slimes against cacti with hoppers in front of the cactus to collect the slimeballs.

Designs
Below is an excellent video example of this farm, including an example world download for further understanding if necessary.