Wombat Behavior
For their size, wombats are remarkably strong and they're extremely efficient excavators. They can reportedly out dig a man with a shovel; they can dig in the hardest of soil, and about the only things that stop them are solid rock or loose sand.

Hairy nosed wombat digging (picture courtesy of Wendy Morphett, used with permission)
Wombats are completely terrestrial. They do not climb trees like their nearest relative, the koala, but they are good swimmers. Young Bare-nosed or Common wombats in foster care apparently do like to climb, however, but Hairy-nosed wombats cannot climb.
Wombats walk with a somewhat awkward, shuffling or waddling gait. Although they appear to be slow and docile, wombats are very alert and can move quickly with great agility when needed.

Bare-nosed Wombat walking (picture courtesy of Womland, used with permission)
Despite their thickset body and stubby legs, wombats can run up to 40 km/hr (25 mph) over short distances. They can cover 100 meters (325 feet) in less than 10 seconds which is as fast as the fastest Olympic sprinters.
While it was once thought that wombats weren't very bright, it's now generally accepted that they are quite intelligent. They are, however, very stubborn and determined, and since they're so strong--built like a tank or bulldozer--they will often go through an obstacle rather than around it. They've been known to go through doors, walls, or anything else that stands in their way. That's probably one of the reasons why they were once considered simpleminded.

Chewbacca, a captive Bare-nosed wombat, likes to play with a swinging log (picture courtesy of Womland)
Playfulness is sometimes cited as a measure of intelligence. Among marsupials, wombats are the most playful. Play will often include "head butting, bitting, and running away to solicit a chase." (ref. 47) When running, a wombat may "indulge in shoulder rolls and somersaults." (ref. 47) Wombats often use biting to express their feelings, usually in a playful manner with no intent to injure.
Wombats have also demonstrated they can be quite clever. In one attempt to trap wombats for scientific research, live traps were positioned at their burrows' entrances. The wombats would usually remain in their burrows for a day or two before either digging around the trap, or digging another exit.

Chewbacca playing with a stump (picture courtesy of Womland)
Wombats seem to be solitary animals and not very social. There appears to be little contact between adults. When wombats meet on the surface, they try to avoid each other.
Bare-nosed or Common wombats do visit each other in their burrows occasionally. In a study of Northern-hairy nosed wombats, they shared a burrow with another wombat 27% of the time. The longest time wombats spend together is when a mother raises her young. The young wombat will leave the mother after about 2 years.

Wombat peeking through fence (picture courtesy of Womland)
Wombats leave scent trails and droppings to mark their feeding grounds. Wombats have unique cube-shaped dung which helps to keep their markers in place. If a new object such as a fallen log is introduced into its home range, a Wombat will mark it with dung. Wombats will also repeatedly rub themselves against certain logs or trees to leave their scent. These objects can often be recognized by their polished surfaces.
Although wombats are naturally very shy, they are inquisitive and have been known to become friendly with people. Wombats are also rather obstinate, and once their mind is made up, will stop at nothing to get what they want.

Southern hairy-nosed wombat sleeping (picture courtesy of Wendy Morphett)
A wombat spends about three-quarters of its time in its burrow. Resting in its sleeping chamber, a wombat can lower its metabolism by about a third, slowing its heartbeat and respiration in order to conserve energy. When relaxed, wombats will often sleep on their backs with their legs in the air. Like many other animals, wombats sometimes snore when sleeping.
Wombats can vary their body temperature which also helps them to conserve energy, but this is only effective up to a temperature of about 25 C (77 F). They do not seem to have sweat glands, and can easily suffer from heat stress.
Bare-nosed or Common wombats have been observed walking into water to soak themselves, apparently in an effort to keep cool.

Wombat eating grass (picture courtesy of Womland)
In the summer, wombats will spend the day in their burrow to avoid the heat. At night, if it's too hot or cool or dry outside, the wombat may remain in its burrow. If the outside air temperature is equal to or lower than their burrow temperature, they'll leave for several hours each night to forage for food alone. The Bare-nosed wombat may travel up to 3 km (1.4 miles) per night and spend 3 to 8 hours grazing.
Bare-nosed wombats may use up to 12 burrows within their territory, with 3 or 4 of these being their main burrows. Each night they will visit up to 4 burrows. More than one wombat will often use the same burrow, but usually at different times. Although they are basically solitary, their territories often overlap. They do seem to have exclusive feeding areas within their home ranges, however.

Southern hairy-nosed wombat on the beach (picture courtesy of Wendy Morphett)
Wombats are nocturnal and are normally most active at dawn and dusk. In the cooler months, they can be active during the day, especially when they'll often lie in the sun to warm themselves.
Southern hairy-nosed wombats were observed to change their patterns of activity depending upon the season. In the summer they were most active in the early morning, and in the winter they were most active in the early evening.
by Peter Marinacci