To address NASA needs for hard X-ray telescopes for starlight detection and wavefront analysis, Physical Optics Corporation (POC) proposes to develop an Adjustable Lobster-Eye X-ray telescope (ALEX) based on POC's Lobster-Eye optics. This approach incorporates two important innovations: (1) new ALEX design--an adjustable array of smaller Lobster-Eye lenses for improved sensitivity, and (2) lens fabrication from two sets of iridium-coated semiconductor-grade silicon plates via POC-patented technology, satisfying the requirement for high flatness compatible with operating wavelengths. These innovations reduce weight 2-3 times compared to conventional Wolter I telescopes, increase angular resolution (8 arcseconds) and versatility, and reduce cost. In high-sensitivity mode, ALEX will have an effective collecting area of >1600 sq. cm. in the 0.25 40 keV range, and operate up to at least 80 keV; in wide-angle mode, ALEX will span a field of view >1.8 deg. In Phase I POC will demonstrate the feasibility of ALEX technology by building/testing a prototype single Lobster-Eye lens, reducing Phase II development risk. In Phase II POC will develop a fully functional prototype lens array and demonstrate its performance as a hard X-ray telescope. These results will allow NASA to build lightweight, high-performance hard X-ray telescopes.