There are currently many libraries spanning most common programming languages. Here you can find a list of these libraries, sorted by programming language.

Cocoa

MGTwitterEngine by Matt Gemmell: Integrate Twitter support into your Cocoa app

Perl

App::Tweet by Joshua McAdams: Tweet on Twitter from the command line

Net::Twitter by Chris Thompson: Perl interface to Twitter.com

Twitter::Shell by Daisuke Maki: Twitter from your shell 

PHP

My Twitter by Andres Scheffer: Set and retrieve a Twitter user\\\\47s status

Twitter by Felix Oghina: Update the status of a Twitter user

TwitterLibPHP by Justin Poliey: Manipulate a Twitter user account

Arc90_Service_Twitter by Matt Williams: Open source Twitter API client

PHP Twitter by Aaron Brazell: Original Code from David Billingham modified and improved

PL/SQL

OraTweet by Noel Portugal: PL/SQL procedures to get friends timeline and post updates

Ruby 

Twitter by John Nunemaker: Command line twits and an api wrapper using hpricot

Twitter4R by Susan Potter: Open-source Ruby library for the Twitter REST API

Twittery by Chris Ledet: Lightweight class for Twitter\\47s API

Python

python-twitter by DeWitt Clinton. This library provides a pure python interface for the Twitter API.

python-twyt by Andrew Price. BSD licensed Twitter API interface library and command line client.