This paper describes the design and implementation of a concurrent compacting garbage collector for languages that distinguish mutable data from immutable data (e.g., ML) as well for languages that manipulate only immutable data (e.g., pure functional languages such as Haskell). The collector runs on shared-memory parallel computers and requires minimal mutator/collector synchronization. No special hardware or operating system support is required. Our concurrent collector derives from sequential semi-space copying collectors. The collector requires that a heap object includes forwarding pointer in addition to its data fields. An access to an immutable object can be satisfied either by the original or the forwarded copy of the object. A mutable object is always accessed through the forwarded copy, if one exists. Measurements of this collector in a Standard ML compiler on a shared-memory computer indicate that it eliminates perceptible garbage-collection pauses by reclaiming storage in parallel with the computation proper. All observed pause times are less than 20 milliseconds. We describe extensions for the concurrent collection of multiple mutator threads and refinements to our design that can improve its efficiency.