You're probably growing one of two species of agapanthus: one is an evergreen and the other is deciduous. The type that's also called the African lily (Agapanthus africanus) is an evergreen that can withstand some frost, especially with protection. It grows in U.S. Department of Agriculture plant hardiness zones 8 through 10. The plant, sometimes called a Headbourne hybrid (Agapanthus campanulatus), is more hardy, grows in USDA zones 6 through 10 and usually loses its leaves in winter.
Both deciduous and evergreen lily of the Nile plants become dormant when weather cools in late fall or winter. In early spring, usually late March or early April, the plant begins to show new growth, with young, strappy leaves appearing at the center of the plant. The exact time when the plant resumes its growth can vary and depends to some extent on how quickly the air and soil warm in that particular spring. This is a good time to cut back any yellowed or cold-damaged leaves on evergreen plants and to remove the remnants of the previous season's leaves from deciduous plants.
Although lily of the Nile usually comes through winter well, you can help protect it from cold by mulching the plant to help keep its roots warm. Apply a 4- to 6-inch-thick layer of organic material, such as straw or shredded bark, around the plant's base in fall. Winter winds might disturb the mulch layer, so renew it as needed. You can also help protect the plant from cold by siting it near a west- or south-facing wall of a building, or planting it on a hillside where cold air drains away to lower parts of the slope.
Keeping a lily of the Nile plant well-watered as winter approaches also helps keep the plant healthy and prepared for cold weather. Once winter arrives, reduce watering while the plant is dormant. Lily of the Nile thrives in any type of garden soil that's well-draining and can do poorly in a spot that tends to stay wet and soggy. If your soil is rich in clay, adding some sand at planting can improve its drainage. This plant blooms best when fed every week or two during the growing season with a balanced, 10-10-10 fertilizer, diluted at a rate of 1/4 teaspoon per 1 gallon of water.