This workshop on Mathematical Foundations of Open Systems explores new research directions towards a logical/mathematical foundation for modeling the behavior of dynamic open systems that evolve over time through self-organization, regulation, and adaptation to changing environments and structures. Such a framework should provide a unified approach for obtaining an advanced understanding of natural systems, the ability to fix and modify them, and to design cyber-physical systems (CPS) in principled ways using new notions of control and coordination. The workshop, held May 23-25, 2010, Philadelphia, PA, is supported by the NSF and other agency members of the interagency coordinating group on High Confidence Software and Systems.