The remotely powered underwater acoustic sensor networks (RPUASN) paradigm is introduced, whereby sensor nodes harvest and store the power supplied by an external acoustic source, indefinitely extending their lifetime. Necessary source characteristics are determined. Feasibility is illustrated with realistic examples, and open research issues are pointed out. Performance of RPUASN is directly related to the sensing coverage and communication connectivity over the field the sensor nodes are deployed. The required number of RPUASN nodes and the volume which is guaranteed to be covered by the nodes are analyzed in terms of electrical power, range, directivity, and transmission frequency of the external acoustic source, and node power requirements.