In MLB, the career leader for innings pitched is Cy Young who pitched 7354 innings in a career that spanned 1890-1911. As far as the 'modern' innings pitched leader, that would be Phil Niekro who pitched 5404 innings in a career that spanned 1964-1987.
Click on the 'Scoreless Innings Leaders' link on this page to see a list of MLB career leaders of scoreless innings pitched.
In MLB, 240 by Justin Verlander of the Detroit Tigers.
That would be Will White of the 1879 Cincinnati Reds who pitched in 680 innings. Since 1900, the most innings pitched in a season is 464 by Ed Walsh of the Chicago White Sox in 1908. Since 1950, the most innings pitched in a season is 376 2/3 by Wilbur Wood of the Chicago White Sox in 1972.
In MLB history, that would be 26 innings pitched by Leon Cadore of the Brooklyn Robins and Joe Oeschger of the Boston Braves on May 1, 1920. The game ended in a 1-1 tie (called due to darkness) and is the longest game, innings wise, in MLB history.
Brad Ziegler Oakland Athletics 2008 32 innings still going
As of the 2007 postseason, the record is held by John Smoltz who has pitched 207 innings in the postseason.
The most innings pitched by a pitcher in a single game is 26 and is held by Leon Cadore of the Brooklyn Dodgers and Joe Oeschger of the Boston Braves on May 1, 1920. The game went 26 innings and wound up in a 1-1 tie. Both Cadore and Oeschger pitched the entire 26 innings. It is the longest game, innings wise, in MLB history.