Grid computing is the use of large collections of heterogeneous, distributed resources (including machines, databases, devices, and users) to support large-scale computations and wide-area data access. The Legion system is an implementation of a software architecture for grid computing. The basic philosophy underlying this architecture is the presentation of all grid resources as components of a single, seamless, virtual machine. Legion's architecture was designed to address the challenges of using and managing wide-area resources. Features of the architecture include: global, shared namespaces; support for heterogeneity; security; wide-area data sharing; wide-area parallel processing; application-adjustable fault tolerance; efficient scheduling and comprehensive resource management. We present the core design of the Legion architecture, with focus on the critical issues of extensibility and site autonomy. Grid systems software must be extensible because no static set of system-level decisions can meet all of the diverse, often conflicting, requirements of present and future user communities, nor take best advantage of unanticipated future hardware advances. Grid systems software must also support complete site autonomy, as resource owners will not turn control of their resources over to a dictatorial system.