Liriope, also commonly known as 'Monkey Grass', makes for a great border though can be used as ground cover.
Pretty, lavender to pink flower spikes emerge and rise above the foliage in summer. Once established, Liriope is virtually indestructible.
Many varieties are available however the one's we plant most are 'Big Blue', which prefers shade or afternoon shade, and 'Variegata', which prefers sun. These two varieties will not run all over your beds or out into the lawn. They stay in neat, mounded clumps that gradually expand in size over the years.
'Silver Dragon', on the other hand, is a spreader. Which can be planted in areas where it will be contained on all sides by concrete driveways, walks or something that will contain it.
When it comes to ground cover, these little ornamental grasses are as tough as they come. All are exceptionally drought tolerant and not picky when it comes to soil.