This paper reports on the development of a new kind of unmanned underwater vehicle which draws inspiration from cephalopods both in terms of morphology and swimming routine. The robot developed here is the first in its kind, being a soft aquatic robot which travels in water by pulsed-jet propulsion. The general design principles of this innovative kind of underwater robot are illustrated and a first prototype is built and tested. The experiments demonstrate an inverse correlation between the frequency of pulsation and the speed of the robot. A mathematical model which associates the kinematics of the pulsating routine to the dynamics of the swimming is devised and compared with the experiments in order to better investigate the interplay of the various design parameters.