Non-Attack interaction
Interaction between players in Dominion comes about through different mechanics. The primary ones are Attack cards and the Supply -- the cards there are accessible to all players enabling players to deny others access to cards by emptying a pile, and to exert influence over the end of the game which affects both players. The effect of denial can be more pronounced if fewer copies of certain cards are available - especially for split piles like Knights and Encampment - and also extends to non-supply cards like Travellers and Rewards. Finally, this effect also appears with respect to the Black Market deck as well as Boons and Hexes.
Other than these core mechanics, there are other effects that cause interaction between players, including some that affect all players equally (like Family of Inventors), some that give minor benefits to other players (like Council Room), some that concern resources that can be fought over (like Artifacts and many Landmarks), and Reactions which can often be used in reaction to another player's doings.
The distinction between Attack-based and non-Attack interaction is somewhat fluid: On the one hand, there are effects that closely resemble attacks but are not (like Haunted Castle), on the other hand, some Attack cards also have interactive effects beside their primary Attack effect, for example Soothsayer (giving a minor bonus to another player), Rogue (interacting with the trash that is accessible to both players), and Black Cat which reacts to another player gaining a card to attack them.
Non-Attack interactive cards
Cards in italics have been removed.
Cards that give a small benefit to other players
Cards that allow another player to make a decision
Effects that apply to all players
Some cards have extra effects depending on the state of the supply which are sometimes highly important (such as with City). Emptying supply piles thereby affects all players when they have access to these effects.
- Poacher
- City
- Animal Fair
- Paddock
- Carpenter
- Territory
- Search
- Tower
- Command variants that play cards from the supply (like Band of Misfits) also depend on the supply because they can't play cards from empty piles.
- Salt the Earth can influence the supply without a player gaining a card.
Similarly, the effects of Forager, Necromancer, and Mountain Shrine depend on cards in the trash, so trashing cards affects all players, if those effects are available.
- Embargo puts an Embargo Token on a supply pile, causing all future buys from that pile to come with a Curse.
- Trade Route produces +
based on the number of Victory card piles that have been gained from. - Tax puts Debt tokens on a supply pile that affect whichever player is next to buy from that pile.
- Family of Inventors makes a non-Victory Supply pile cheaper for all players
- Messenger can give out a useful or harmful card but all players gain a copy of the card.
- Prophecies affect all players when they are activated.
Resources that can be fought over or taken away
Some effects can gain cards from the trash which is accessible to all players, so you can take cards from there for yourself and deny them to your opponent.
Gathering cards can collect 
tokens over time which every player has the opportunity to take away:
Some Landmarks come with a limited amount of tokens that can be fought over:
Some cards offer access to a unique instance of an effect - mostly Artifacts - which can be passed back and forth between players:
- Fool with Lost in the Woods
- Border Guard with Horn and Lantern
- Flag Bearer with Flag
- Swashbuckler with Treasure Chest
- Treasurer with Key
Cards whose effects depend on another player's deck, hand, or play area
Interactions with other players gaining cards
- Monkey
- Smugglers
- Pirate
- Fool's Gold
- Treasure Hunter
- Road Network
- Falconer
- Invest
- Cutthroat is an Attack, but it has an interactive ability unrelated to its status as an Attack.
- Mapmaker
- Stowaway
Reactions
Most Reactions can trigger on the doings of another player.
Attack-like effects
These are not attacks but have interactive effects that are typically associated with attacks:
- Ill-Gotten Gains
- Noble Brigand: Although an Attack when played, the when-buy effect is not technically an Attack and as such cannot be protected against by cards like Moat or Lighthouse.
- Raid
- Haunted Castle
- Circle of Witches
- Masquerade: each player passes a card to the next player.
- Maelstrom: each player trashes from their hand, which may be beneficial or harmful
Possession is also often considered to be similar to an Attack.
Gallery
Cards that give a small benefit to other players
Cards that allow another player to make a decision
Effects that apply to all players
Resources that can be fought over or taken away
Cards whose effects depend on another player's deck or hand
Interactions with other players gaining cards
Attack-like effects
Removed
Trivia
Dominion is occasionally caricatured as "multi-player solitaire", in that in some circumstances it is possible for each player to independently build their own deck, with no direct interaction between different players' strategies and deck compositions. This tendency is mitigated, however, by the presence of cards that affect players other than their owners; this arguably makes the game more fun and strategic by keeping each player involved even during other players' turns. Attack cards are interactive in this way, but also tend to delay the game and are unpopular with some players; for this reason Donald X. Vaccarino thought it important to include non-Attack cards that cause interaction between players as well. At an early stage of development non-Attack interaction was the theme of the "War" expansion that eventually was revised into Dark Ages; it ended up being a subtheme of Prosperity, an expansion with relatively few Attack cards. But some such cards are present in each expansion.