Terraced gardens date back many centuries in mountainous areas, such as Peru, where the Inca Indians are believed to have developed this method. Like them, and Asian farmers who grow rice in terraced paddies, you can carve “stairs” on your hillside, creating flat spaces where you plant your lawn, trees, vegetables, ornamentals and other plants. Terraces prevent erosion and rapid water runoff, and they also look attractive.
Raised beds are a sensible method of landscaping on hillsides. Not only do they provide flat areas where grass and other plants can grow, they also can contain highly nutritious soil components when homeowners build them with compost, peat moss, leaf litter and other organic materials. When you build raised beds on a hillside, encircle them with rocks or bricks to prevent the soil and other materials you include from washing downhill. Plant some beds with grass and others with flowers, vegetables, herbs, shrubs and trees, depending on your taste and climate zone.
Certain grasses prevent erosion and look attractive. Vetiver grass has both of these benefits and also repels termites due to fragrant oils in its roots. Although the root system does not cause this grass to become invasive, vetiver grass forms dense clumps that hold soil in place and greatly limit erosion. Vetiver is often grown in hedges, which you might intersperse with terraces and raised beds to effectively increase soil stability.
Mulch and trees of many types can help to prevent erosion in hilly areas. The large root systems of trees become securely anchored in the soil, thereby stabilizing it and preventing it from washing away, as often happens on hills after forest fires or heavy rains denude an area. In remote areas, a practice known as "aerial mulching" is used after fires and floods to prevent erosion. Homeowners can use a scaled-down version of this practice by thickly spreading straw or other types of mulch in such areas to effectively curtail erosion. Later, when you plant your lawn, it adds more stability to the soil and further prevents erosion.