Usually used in reference to a computer application, especially a computer game, a text-based application is one whose primary input and output are based on text rather than graphics. This does not mean that text-based applications do not have graphics, just that the graphics are secondary to the text.
For instance, a game such as Triple Play, a graphics-based baseball game, allows users to manipulate graphical elements in an attempt to achieve particular outcomes in a simulated baseball game. A text-based game like Out of the Park Baseball, on the other hand, allows users to manipulate strategies and text elements to achieve particular outcomes. Any graphics shown are secondary to and supportive of this.
Examples of text-based games include Zork, Cantr II, Championship Manager, Fantasy Empires, and Hollywood Mogul.
More on [ Text-based ]

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A Matter of Choice - Some of the pros, cons, irritations, and joys of using a text-based web browser, from one person's point of view.
Archives of the ACCMAIL Mailing List - ACCMAIL is a moderated forum for news, comments, questions and answers on how to browse the World-Wide Web, retrieve files from FTP sites, and read and post to Usenet newsgroups - using email only.
ELinks - ELinks is an open-source (GNU) text WWW browser supporting almost full HTML/4.0 (without CSS; including tables, frames and full color support), HTTP/1.1, FTP, tabs support etc. It is a full-featured browser with focus on menu-driven interface and high configurability.
Meta Description: [ Home of the ELinks project ]
EnterWEB - 3270 terminal web browser for mainframes running OS/390.
Links - Links is a WWW text browser with table support.
Links List Info Page - Mailing list for Links, a text based web browser.
Links@Sourceforge.net - Links is a text-based browser with support for HTML tables and frames. For Unix, OS/2, BeOS, MacOSX, Win32 (Beta).
REVOBILD - Pure text Internet tools for the DOS operating system, including a web page grabber and mail reader.
w3browse - A text-based web-browser and e-mail client. It has been designed to be mostly platform independent and is able to run on a wide variety of systems including UNIX/Linux and MS-Windows.
Meta Description: [ w3browse is a text-based web-browser and e-mail-client. Designed to be mostly platform independent, it is able to run on a wide variety of systems including UNIX/Linux and MS-Windows. ]
Wanna-Be - Limited text-only web browser for MacOS under development by David T. Pierson.
www4mail - An open source application that allows the user to navigate off-line and search the whole Internet via electronic mail (e-mail) by using any standard Web browser and a MIME (Multipurpose Internet Mail Exchange) aware e-mail program.
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