Each opponent can't draw more than one card each turn.
Whenever you or a permanent you control becomes the target of a spell or ability an opponent controls, you may draw a card.
8/23/2016 Your opponents can draw a maximum of one card on each player’s turn. Subsequent card draws are ignored.
8/23/2016 If an opponent hasn’t drawn any cards in a turn, and a spell or ability instructs a player to draw multiple cards, that player will just draw one card. However, if the draws are an optional cost to generate an additional effect (“[That player] may draw two cards. If they do, [effect]” or similar), the player can’t pay that cost in full and can’t draw any cards.
8/23/2016 Leovold will “see” cards drawn by opponents earlier in the turn on the turn it enters the battlefield, although Leovold can’t affect cards drawn before it entered the battlefield. For example, if an opponent draws two cards, then Leovold enters the battlefield, that opponent can’t draw more cards that turn, but the two drawn cards are unaffected.
8/23/2016 Replacement effects can’t be used to replace draws that Leovold disallows.
8/23/2016 If you and a permanent you control each become the target of the same spell or ability an opponent controls, Leovold’s ability will trigger twice.