The C programming language is at least as well known for its absence of spatial memory safety guarantees (i.e., lack of bounds checking) as it is for its high performance. C's unchecked pointer arithmetic and array indexing allow simple programming mistakes to lead to erroneous executions, silent data corruption, and security vulnerabilities. Many prior proposals have tackled enforcing spatial safety in C programs by checking pointer and array accesses. However, existing software-only proposals have significant drawbacks that may prevent wide adoption, including: unacceptably high runtime overheads, lack of completeness, incompatible pointer representations, or need for non-trivial changes to existing C source code and compiler infrastructure. Inspired by the promise of these software-only approaches, this paper proposes a hardware bounded pointer architectural primitive that supports cooperative hardware/software enforcement of spatial memory safety for C programs. This bounded pointer is a new hardware primitive datatype for pointers that leaves the standard C pointer representation intact, but augments it with bounds information maintained separately and invisibly by the hardware. The bounds are initialized by the software, and they are then propagated and enforced transparently by the hardware, which automatically checks a pointer's bounds before it is dereferenced. One mode of use requires instrumenting only malloc, which enables enforcement of per-allocation spatial safety for heap-allocated objects for existing binaries. When combined with simple intra-procedural compiler instrumentation, hardware bounded pointers enable a low-overhead approach for enforcing complete spatial memory safety in unmodified C programs.