What Anglo-Saxon beliefs are depicted in Beowulf?

Beowulf demonstrates a number of Anglo-Saxon beliefs which fall within the so-called "heroic code" of warriors. One of these is the idea that one must stay with his lord to the end—as demonstrated in the "hero on the beach" scene where Beowulf dies and gives his last speech to Wiglaf. Another is the concept of the heroic boast: it was considered customary for warriors to detail their feats to each other, illustrated in the scene in the hall when Beowulf describes his previous encounter with a sea-monster. This was not seen as being arrogant, as it might be today, but instead was an indication of a warrior's bravery.

We can also see the strong belief that life was only worth living as part of a tribe or clan, which must therefore be defended to the death. Grendel, the monster, has been exiled from any clan: therefore, he envies those who are part of the group and attacks them one by one while they sleep. This idea is reflected in Anglo-Saxon elegies such as "The Wanderer" and "The Seafarer" where the pain suffered by the speaker comes from having been cut away from a group. It is interesting to compare these speakers to Grendel.

The Anglo-Saxons also believed in the transitory nature of existence and of physical things; they were very focused on the idea of things being "laene," or transitory. At the end of the poem, when Beowulf is killed, he is buried with his treasure, which is now "as useless to men as it ever was." This underscores the idea that material things are ultimately not of great import, even though kinship bonds were often symbolized through the exchange of rings and other jewelry.

The role of women in Anglo-Saxon society is also symbolized through the behavior of good queens like Wealtheow, a peaceweaver, and equally by bad queens like Modthryth, who does not perform this duty of creating unity between tribes.

What three values or ideals from pre-Christian Anglo-Saxon culture does Beowulf exemplify?

Beowulf is the ideal Anglo-Saxon hero. He is physically strong and a cunning warrior, for one thing. During his three battles, he overcomes his increasingly stronger adversaries. He overcomes Grendel through brute force, but he overcomes Grendel's mother through cunning, grabbing onto a magic sword to strike her down right before she is able to kill him first.

Anglo-Saxons seek immortality through becoming famous. Beowulf does not speak much about an afterlife where he will be rewarded or punished. Rather, he talks about wanting to become famous by fighting monsters so his name will be remembered long after his death. Hrothgar tells Beowulf,

Thy fame is extended through far-away countries,
Good friend Beowulf, o’er all of the races,
Thou holdest all firmly, hero-like strength with Prudence of spirit . . .

This shows how important reputation is to the Anglo-Saxon people. This is the fool-proof way to make sure one's memory lives on after death. When Beowulf is buried at the end of the poem, the love of his people is the greatest legacy he leaves behind. Even though he is gone and can no longer protect them, his memory will live forever.

The Anglo-Saxons also prize loyalty to a king. We see this with how Wiglaf remains loyal to Beowulf even in the face of fiery death at the hands of the dragon. The other warriors abandon their king, which is shameful. A good warrior is always loyal to his king, even unto death. The king, in turn, is generous with his warriors, letting them make merry in the mead hall and sharing treasures with them after battle.

What characteristics do the Anglo-Saxons value most in Beowulf?

Beowulf certainly embodies what the Anglo-Saxons valued in a hero. A courageous, skillful, and imposing warrior attractive in appearance, Beowulf hears of Hrothgar's trouble with Grendel and resolves to journey to the Danish land to help King Hrothgar and his people. His determination to help others in distress, risking his own life to do so, is another valued characteristic.

When he heard about Grendel, Hygelac’s thane
was on home ground, over in Geatland.
There was no one else like him alive.
In his day, he was the mightiest man on earth,
high-born and powerful. He order a boat
that would ply the waves. He announced his plan:
to sail the swan’s road and search out that king,
the famous prince who needed defenders.

Beowulf is considered most powerful: his appearance is described by one of Hrothgar's watchmen as the Geats land their boat on the Danish shore:

Nor have I seen
a mightier man-at-arms on this earth
than the one standing here: unless I am mistaken,
he is truly noble.

Beowulf does not shy from telling about his own brave feats at the welcome feast in Hrothgar’s hall. In fact, formal boasting of accomplishments in fight or battle was greatly admired. When challenged by the jealous Dane Unferth, Beowulf describes how he swam in full armor for five nights and fought off nine sea monsters.

That night, Grendel attacks the hall where Beowulf lies awake while his men sleep. Beowulf has vowed to fight the monster without weapons, showing his incredible courage and willingness to die in the fight if necessary.

Grendel discover[s]

himself in a handgrip harder than anything
he had ever encountered in any man
on the face of the earth.

Beowulf has fiercely grabbed Grendel’s arm and hand--they fight violently, knocking over and smashing benches as Grendel struggles to free himself. The result of the combat is that Beowulf entirely rips off Grendel’s arm and shoulder; the monster flees to die in his den. It turns out that human weapons cannot harm the monster, so Beowulf’s decision to fight unarmed was a wise one. Thus Beowulf demonstrates his incredible bravery, his unmatched strength, and his keen wisdom in his defeat of Grendel.

The Geat captain
had boldly fulfilled his boast to the Danes...
Clear proof of this
could be seen in the hand the hero displayed
high up near the roof: the whole of Grendel’s
shoulder and arm, his awesome grasp.

Anonymous. Beowulf. A New Verse Translation by Seamus Heaney. New York: Farrar, Straus and Giroux, 2000. Print.

What virtues did the Anglo-Saxons value, based on the beginning of Beowulf?

Anglo-Saxon’s considered strength, bravery and battle-sense virtuous traits.

Wulfgar, who is himself “well-known to many for his might of mind, courage, and wisdom” (ch 5), is clearly impressed by Beowulf.  He seems humble and honest, and he can tell by the armor that his people are warriors.

“Men from afar have come hither over the ocean's paths—people of the Geats—and the most noble of their band is named Beowulf. … By their war-gear they appear worthy warriors, and their leader, a hero who led his band hither, is surely a valiant man.” (ch 5)

Since the poem makes a point to emphasize bravery and strength, we can see that these are thing the Anglo-Saxons valued.  Beowulf is considered remarkable because he can fight a monster without a sword.  He is honest, because he explains that the rumors about him killing his friend when he was younger are not true.  He behaves properly and humbly before the king, and that also endears him.