Japanese honeysuckle was introduced to North America sometime in the early 1800's, most likely imported as an ornamental specimen and horticultural groundcover. It is certainly well-suited for use as a groundcover, as the vines can extend more than 30 feet in length (isolated sources say it can attain up to 80 feet in length), and can both spread along the ground, and also climb trees, plants, and structures. Vines stretching along the ground send down roots at each nodule that contacts the soil, creating a dense mat of vegetation. The vine grows vigorously in sun and more slowly in shade, and spreads both vegetatively, by rooting along the length of the vine, and by seed dispersal. In my own experience, I have had Japanese honeysuckle emerge in three different areas of my yard in central Illinois, most likely from seeds dropped by birds.
In the mid-1900's, Japanese honeysuckle was widely planted as a groundcover to stabilize road banks, provide erosion control, and provide food and cover for wildlife. Unlike many invasive plants, it does serve as an viable food source and shelter for many animals, and may make up a significant portion of the diet of white tailed deer in the southern region of the United States.
The impact on native plants, however, has been devastating. In many areas, Japanese honeysuckle is now the dominant groundcover in wooded and forest areas, far outgrowing the native species. "This aggressive vine seriously alters or destroys the understory and herbaceous layers of the communities it invades, including prairies, barrens, glades, flatwoods, savannas, floodplain and upland forests. It may become established in forested natural areas when openings are created from treefalls or when natural features allow a greater light intensity in the understory. Japanese honeysuckle also may alter understory bird populations in forest communities."  It rapidly twines up every vertical surface, girdling small saplings, covering shrubs and small trees so thoroughly that they collapse under the weight of the vine. It also forms such a dense canopy of leaves that it almost completely blocks off light to the ground below, killing off the native vegetation that would normally grow in the area.
The extent of the spread throughout the United States can be seen in this map, from the University of Georgia Center for Invasive Species and Ecosystem Health:
Japanese honeysuckle is present at some level in all but 7 states (OR, ID, MT, WY, ND, SD, and AK), though the severity of the problem is significantly worse in the states in the southeastern region, with many states reporting its presence in every county. Bugwood.org lists Japanese honeysuckle as the "[m]ost commonly occurring invasive plant in the South, overwhelming and replacing native flora in all forest types over a wide range of sites or occurring as scattered plants."  When I drove through Iowa, Illinois, Indiana, and Kentucky this spring, it was visibly prevalent, often making up a significant portion of the greenery along roadsides and ditches. It is easily identified, even from a distance, during its bloom season in the spring, when it is covered in white and yellow blooms.
As a semi-evergreen plant, this vine also enjoys a prolonged growing season. While native plants are still in their winter dormancy, the Japanese honeysuckle is in rapid growth, especially in areas of full sun. Likewise, when native plants become dormant in the fall with the approach of cold weather, the honeysuckle continues to grow, forming ever denser mats that prevent the germination or survival of native tree and plant seedlings the following spring. Its foliage maintains its ability to photosynthesize even during the winter months, giving it as much as two months headstart on the native deciduous trees.
These dense mats and climbing vines may pose another danger, a potential increase in forest fire severity. Since the vines frequently grow to great heights, reaching well into the canopy of the trees that support them, there is a demonstrated risk that they will act as fire ladders, allowing wildfires to reach into the canopies in wooded areas, instead of quickly burning through the understory.
In this instance, we can take advantage of the growth patterns of Japanese honeysuckle to aid in its removal. Since it is semi-evergreen, and leafs out earlier than most native species, it is easy to find and identify in late winter and early spring. This would be an ideal time to cut, pull, or dig the vines. It is possible to manually remove the roots of immature specimens, especially in moist soil, as the roots of young plants are not extensive or deep. The easiest time to remove a plant is when it is still young, and hasn't produced fruits and seeds. Well-established thickets are more difficult to fully eradicate, though thankfully the seeds don't seem viable over a particularly long period.
Damage to the cambium layer within the stems actually triggers increased sprouting. For this reason, simply cutting back the plant, or allowing intermittent browsing by goats or other livestock, will not provide long-term control, and may actually promote more rapid and aggressive growth. Likewise, intermittent mowing is not recommended, as the roots will merely send up multiple new stems and perhaps increase the density of the infestation. Isolated or infrequent burning has not proven to be an long-term deterrent for Japanese honeysuckle, as it often returns with renewed vigor, and re-establishes mature growth more quickly than the native plants you wish to re-colonize the area.
Chemical treatments are more effective, if the infestation is too extensive for manual removal. This is another occasion in which the growth habits of the plants can work to our advantage. The Missouri Department of Conservation recommends treatment with glyphosate herbicide (tradename Roundup) or Crossbow (triclopyr and 2,4-D) shortly before the first hard freeze in the autumn, once the native vegetation in the area has become dormant. It is a balancing act, between providing thorough coverage on as much of the honeysuckle foliage as possible, while minimizing contact and dripping onto the native plants you wish to preserve.
For vines that are entwined around trees or other plants, you may cut the vines and immediately treat the cut surface with your chosen herbicide, again minimizing contact with the supporting plant.
 Entry on Missouri Department of Conservation, Problem Plants: http://mdc.mo.gov/your-property/problem-plants-and-animals/invasive-plants/japanese-honeysuckle-control
 Wiki entry on Bugwood.org, developed by the Center for Invasive Species and Ecosystem Health at the University of Georgia: http://wiki.bugwood.org/Archive:IPSF/Lonicera_japonica
National Park Service Integrated Pest Management Manual: http://www.nature.nps.gov/biology/ipm/manual/exweeds2.cfm
National Park Service entry on Japanese Honeysuckle: http://www.nps.gov/plants/alien/pubs/midatlantic/loja.htm
Thumbnail Closeup of Honeysuckle blossom: Flickr Creative Commons, submitted by MomentsforZen. Some Rights Reserved.
Shed overgrown with Honeysuckle: From Invasive.org, submitted by Chuck Bargaron, University of Georgia.
Map of Honeysuckle Prevalence: From EDDMapS. 2014. Early Detection & Distribution Mapping System. The University of Georgia - Center for Invasive Species and Ecosystem Health. Available online at http://www.eddmaps.org/; last accessed August 16, 2014.
Honeysuckle vines and leaves: Dave's Garden PlantFiles, submitted by Dave's Garden member Gabrielle. Used with permission. All Rights Reserved.
Honeysuckle fruits: Dave's Garden PlantFiles, submitted by Dave's Garden member Hilary Nightingale (philomel). Used with permission. All Rights Reserved.
Blooming honeysuckle plant: Flickr Creative Commons, submitted by David Heise. Some Rights Reserved.