Asgard shields are the primary defensive technology used by Asgard and Tau'ri spaceships, and are among the most advanced shields known to exist. The technology was shared by the Asgard with the Tau'ri in thanks for their contributions in defeating the Replicators, giving Tau'ri ships a great advantage over the other space-faring races of the universe.
Asgard shields create an elliptical "bubble" around the spaceship they are equipped to, which absorbs and diffuses the energy from weapons fire. The shield matrix will gradually degrade under fire (rather quickly under sustained bombardment, but only if the shield is attacked by powerful weapons), resulting in more and more energy from attacks bleeding through and causing damage to the ship. However, as pointed out, the weapons impacting the shield must be quite powerful, otherwise the shields will hold even under sustained fire. Lesser weaponry like the one on a Ha'tak vessel have little to no effect on the shield even if multiple vessels are bombarding it simultaneously. Additional energy can be allocated to the shield matrix to reinforce it, but the only way for a ship to ultimately survive a continued assault is to withdraw and give the generators time to recharge and replenish the matrix. (ATL: "The Siege, Part 3", "No Man's Land")
Asgard shields far outstrip the much more primitive Goa'uld shields in effectiveness. Despite this, they become considerably less effective when facing foes of a comparable technological level. The exact strength of an Asgard shield, like most Asgard technology, appears to be relative to the amount of power channelled into them, meaning the shields on Asgard ships are much more powerful than those of Tau'ri spaceships unless a Zero Point Module or equivalent power source is interfaced, which renders them even more powerful than the shields featured on Asgard vessels. The most advanced shields ever developed by the Asgard were equipped on the Odyssey before they committed mass suicide, allowing the ship to survive sustained fire from the central weapons of multiple Ori warships for several minutes, a most incredible feat currently unmatched by the shield technology of all other races, including Ancient shields. (SG1: "Unending", "The Ark of Truth") With a ZPM powering the shield it can withstand virtually anything.
Even without a ZPM backing them, Asgard shields are capable of directly resisting Drone weapons; however Todd remarked that it would only take one of Atlantis's drones to disable the Daedalus' hyperdrive, implying that the drones on Atlantis are somehow able to penetrate Asgard shields without much resistance (although he could have been underestimating the shield's capabilities, may not had sufficient control of systems to raise them, or he may have assumed the Atlantis expedition knew enough about the Asgard shields to get a drone to bypass them completely. It should also be noted that the Daedalus was at the time severely damaged by sabotage - Ronon blew up several key systems in an attempt to stop the Wraith. (ATL: "Be All My Sins Remember'd", "The Lost Tribe")