Different varieties of Agapanthus vary to the extent they will tolerate frost and cold. For best results, it is important to select the most suitable variety for your garden and to select the right planting place. Like so many of our garden favourites, when growing Agapanthus, the trick is to get the "right plant in the right place" and this is very much the case with Agapanthus. As a rule of thumb, the deciduous varieties are more hardy than the evergreen varieties, and the less hardy types will need a winter mulch and frost protection.
In more exposed gardens a fully hardy variety, H4 which is hardy down to a range of -5-10 would be the better option and this group includes readily available varieties such as A. Blue giant, A. Midnight blue, A. Lilliput A, Silver Moon, (as you may expect a short variety up to 10cms) A. Snowy Owl with white flowers. Check when you buy, either the label or website will state if the variety is fully frost hardy. Selecting a hardy variety is essential unless your garden is sheltered, or you can bring the plants under glass protection for the winter. More about frost hardy.
If your garden is exposed,subject to winter wet, and/or prone to lots of frost, or you prefer to grow a more tender variety, grow Agapanthus in pots and move into an unheated conservatory or greenhouse for the winter. There is a view that Agapanthus flowers better in containers, and you will notice visiting open gardens, noticeably further north. Agapanthus are frequently grown in containers. In colder but not too exposed areas, Agapanthus will survive a winter if covered in a good mulch. In colder areas, Agapanthus will need shelter and winter protection.
My garden is exposed, and wet, and I grow Agapanthus in containers. Usually I bring them into the greenhouse over winter, more than once I have left outside with a thick mulch, but that is a risk if the roots freeze, the plant will die.
It can take a little time to get Agapanthus to flower, although once established, they will often flower for some years with little trouble. If you live in Cornwall, or the Scilly Isles, you are lucky, and no doubt laughing up your sleeve at this advice, which you can ignore. In these areas, Agapanthus grows and flowers abundantly in the milder conditions. ( Which may explain why in New Zealand it's status is of an invasive weed)