We have developed a strain sensor using highly densely-packed, oriented single-walled carbon nanotube (CNT) films bonded to a stretchable polymer substrate. The sensor measures strain by detecting changes in the electrical resistance of the films. This CNT strain sensor can detect strains of up to 280 %, about 50-times greater than conventional metal strain sensors. In addition, durability was demonstrated by repeated application of 150 % strain over 10,000 cycles, and strain response time was found to be 14 ms. The sensor is less prone to creep than strain sensors made of conducting polymeric composites and is more than 20-times faster in creep recovery. The CNT strain sensor can be easily attached to clothing or directly to the body to monitor motion from knees to fingers. It is likely to be used in future wearable devices such as in recreational and medical applications.