This paper presents a novel Shape Memory Alloy (SMA) linear actuator and its application driving the joints of an anthropomorphic robotic hand. Its main feature is that it is a flexible SMA actuator, it is designed so that it can be bent. The possibility of bending the actuator while preserving its capacity to produce a linear movement allows to use very long SMA wires, able to yield a great linear displacement, in a confined space. This feature provides better integration and adaptability than other existing high-displacement SMA actuators. The mechanical design of the presented SMA flexible actuator is based on the Bowden cable transmission system, using a multilayer sheath with a PTFE inner sheath to reduce friction losses and a stainless steel outer sheath that functions as a heat sink to reduce the cooling time of the SMA element. To test the SMA actuator with a real device, a robotic hand has been built using low cost rapid prototyping techniques from an open source design. The designed flexible SMA actuator has a great potential to be used in humanoid robots or robotic prostheses, due to its adaptability, output force, low weight and silent operation.