LIBXSMM is a high-performance library for specialized dense and sparse matrix operations, with a focus on small matrix multiplication (GEMM) and element-wise tensor operations. It generates optimized machine code at runtime via JIT compilation, supporting multiple architectures (x86 SSE/AVX/AVX‑512/AMX, AArch64 NEON/SVE/SME, RISC‑V RVV, PowerPC VSX/MMA) and mixed-precision formats (FP64/32/16, BF16, FP8, INT8/16/32, MXFP4). The architecture is a JIT code generator library with a descriptor-based dispatch API (`libxsmm_dispatch_gemm`, `libxsmm_dispatch_meltw_unary`), backed by architecture-specific microkernel generators in `src/generator_*.c`. Key entry points include the public header `include/libxsmm.h`, the CPU feature detection (`libxsmm_cpuid_*.c`), and the binary export tool for offline kernel generation. Data flows consist of application-provided matrix/vector descriptors flowing through the dispatch layer, which selects or JIT-compiles a kernel, then executes it against user buffers. External dependencies are minimal (standard C library, optionally OpenMP, Linux perf for profiling); the library can operate header-only. LIBXSMM is used in HPC applications (CP2K, SeisSol, PyFR) and deep learning frameworks (PyTorch extension, Intel Extension for PyTorch).