A rotary steerable apparatus is provided having an actuator for pushing the bit or pointing the bit that includes a shape memory alloy. An elongated form of the alloy, such as a wire or rod, is employed in a mechanism that applies force in a direction transverse to the wellbore in response to a change in length of the alloy. Temperature of the alloy is controlled to change shape and produce the desired force on pads for operating the apparatus. The apparatus may be used with downhole power generation and control electronics to steer a bit, either in response to signals from the surface or from downhole instruments.