The MLB draft has different rules from other professional sports drafts such as the NFL, NBA, or NHL.
There may be up to 50 rounds in the MLB draft. With 30 teams, that could mean 1,500 picks. However, there are more rules complicating that number.
First, depending on how many free agents a team lost (and how good those free agents were), teams might be awarded "compensatory picks" that are sandwiched between the first and second round.
Second, a team can stop picking at any round by "passing." The remaining teams will continue picking.
In the 2008 draft, 1,504 players were drafted. In 2007, 1,453 players.
The MLB draft currently held in the first week in June.
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