Methods and apparatuses relating to stateful compression and decompression operations are described. In one embodiment, hardware processor includes a core to execute a thread and offload at least one of a compression and decompression thread, and a hardware compression and decompression accelerator to execute the at least one of the compression and decompression thread to consume input and generate output data, wherein the hardware compression and decompression accelerator is coupled to a plurality of input buffers to store the input data, a plurality of output buffers to store the output data, an input buffer descriptor array with an entry for each respective input buffer, an input buffer response descriptor array with a corresponding response entry for each respective input buffer, an output buffer descriptor array with an entry for each respective output buffer, and an output buffer response descriptor array with a corresponding response entry for each respective output buffer.