The problems

Reynoutria japonica

Because of their highly competitive nature, invasive exotic plants can sometimes eliminate local species or more often prevent them from growing, by competing effectively with them for light, nutrients*, available space, etc. Some have a direct impact on the environment and manage to radically transform it.
For example, when an amphibious plant such the large-flowered primrose-willow (Ludwigia grandiflora) covers almost an entire lake, the aquatic plants and fauna decrease due to the lack of light, reduced oxygenation, accumulation of organic matter and finally deterioration in water quality. By colonizing riverbank soil rich in water and nitrogen, Himalayan Balsam (Impatiens glandulifera), an annual plant, encourages the undermining and erosion of riverbanks by preventing trees from moving in (alder, willow, ash) which would normally secure and stabilize them. Similarly, a tree such as the Black Locust (Robinia pseudoacacia) enriches the soil it colonizes with nitrogen due to the fixing bacteria in its root system; the many varieties of herbaceous plants specific to poor soil are thus eliminated.

Over and above the ecological impact, invasive plants can seriously affect a variety of human activities: navigation, fishing, water sports, leisure activities and also agriculture, with the loss in value of infested agricultural fields and pastures. 

Finally, the proliferation of certain species poses problems for public health:  contact with the sap of Giant Hogweed (Heracleum mantegazzianum) or the leaves of  Poison ivy (Rhus radicans) causes severe dermatitis and the pollen of Common ragweed (Ambrosia artemisiifolia) is responsible for severe allergies.