The rotation of a screwball must be in the opposite direction of a curveball or a slider. In order for a right-handed pitcher to create this rotation, the hand and arm must pronate, turning the ball moving from the right side of the baseball to the left side (or left to right for a lefty.) The easiest way to achieve this movement is with a "3/4" overhand motion.
There are several popular grips for the screwball, however this is probably the least important aspect. All that is required is that the fingers have a solid grip on the ball and adequately "grab" it during the period of rotation. Since the rotation is caused solely by the arm movement, precision finger placement is not required. That being said, one easy to learn grip would be gripping the ball like a two seam fastball, and then moving the ball deeper into the hand. The pitcher's fingers should now arc over the ball (like it would with a curveball grip), but be directly behind the center of the ball (like a fastball). Putting the ball deep in the hand will not allow for proper "grab" on the ball and will provide maximum rotation (if thrown as suggested). If the ball is gripped with a standard fastball grip, the movement of pitch will be diminished and will be more like a fastball tailing inside to a right handed batter. Such movement may not be desired, depending upon the pitchers goal.
The final moments of the pitch are key to properly executing a sound screwball. The majority of the motion up until this point should be identical to a fastball delivery. The only exception is the position of the pitcher's hand, as the index and middle finger should be located at 2-3 o'clock. At the end of the release the pitcher must follow through in a similar fashion to finishing a curveball, except his hand will be moving down the left side of the ball. During this time, the forearm will also begin to pronate, turning counter-clockwise. A violent and drastic pronation is not required and may potentially be bad for the pitcher's arm. It is only necessary to rotate the forearm enough to move the pitcher's fingers to a 10-11 o'clock position. The actual rotation isn't being generated so much by the turning of the forearm, but rather the downward pull of the arm (similar to the 12-6 curveball). The pronation of the forearm is simply meant to get the fingers to "roll off" the correct side if the ball.
If done properly, the ball will make a 2-3 foot move from left to right. Depending on arm angle, the ball may or may not have a sinking action. At most levels of play this can be an exceptionally difficult pitch to hit due to its degree of rarity. For right handed pitchers, the pitch is used to jam right handed hitters, and thrown outside to left hand hitters, making them chase the ball out of the strike zone.
Those learning the pitch should start out slow and never force their arm into doing something too uncomfortable. If done improperly the pitch can cause various shoulder and wrist injuries.
The difficulty in mastering the screwball, and the unusual stress it can place on the pitching-arm — throwing the screwball requires that the forearm be powerfully pronated as the ball is released — has made the pitch an increasingly-rare part of the modern pitcher's pitching arsenal. Christy Mathewson said of the screwball: "It is a very hard ball to deliver. Pitching it ten or twelve times a game kills my arm, so I save it for the pinches." Carl Hubbell, who threw the screwball much more often than did Mathewson, twisted his left arm so severely from years of throwing the pitch that his left palm eventually faced outward when the arm was at rest. In general, the pitch is seldom recommended to young pitchers because of the potential harm it can do to their arms. 
However, 1974 Cy Young Award winner Mike Marshall, who has a Ph.D. in kinesiology, and who was a renowned screwballer himself, believes that the screwball can be safely thrown by pitchers if the the pitch is properly taught under supervision. 
Thrown by a right-handed pitcher, a screwball breaks from left to right from the point of view of the pitcher; the pitch therefore moves down and in on a right-handed batter and down and away from a left-handed batter. Thrown by a left-handed pitcher, a screwball breaks from right to left, moving down and in on a left-handed batter and down and away from a right-handed batter.
Due to this left to right movement of the ball when thrown by a right-handed pitcher, a screwball is often used by right-handed pitchers against left-handed batters in the same way that a slider is used by right-handed pitchers against right-handed batters.
The first prominent screwball pitcher was Hall of Famer Christy Mathewson, who threw what he termed the "fadeaway". However, Mathewson did not invent it; rather he learned it from teammate Dave Williams, with whom he played on a semi-pro team in Pennsylvania in 1898. Research indicates, however, that Williams did not invent the screwball either; Mickey Welch, a 19th century star, might have thrown a screwball.
Carl Hubbell, probably the most renowned practioner of the screwball in the history of the Major Leagues, was known as the "scroogie king" for his mastery of the pitch and the frequency for which threw it. According to Hubbell, "The screwball's an unnatural pitch. Nature never intended a man to turn his hand like that throwing rocks at a bear." Other notable screwball artists included Cy Young Award winners Mike Cuellar, Willie Hernández, Fernando Valenzuela and Mike Marshall. In addition, Boston Red Sox starter Bill Lee and New York Mets (and later Philadelphia Phillies) reliever Tug McGraw also built successful careers using the screwball. John Franco, longtime Cincinnati Reds and New York Mets reliever, used this pitch often to strikeout right handed batters.
Jim Mecir, who retired from the Florida Marlins following the 2005 season, was one of the last major-leaguers to throw the pitch. His screwball was a function not only of his arm motion, but also of an unorthodox delivery. He was born with two club feet; childhood surgery enabled him to walk but has left him unable to properly push off the pitcher's rubber with his right foot