This paper proposes a stacked intelligent metasurface (SIM)-assisted integrated sensing and communications (ISAC) system, where the SIM is used to enhance both communication and sensing performance. The system aims to estimate the complete target response matrix relative to the SIM while satisfying minimum signal-to-interference-plus-noise ratio (SINR) constraints for communication users (CUs) and maximum transmit power. The paper introduces a multi-layer alternating optimization (MAO) algorithm to jointly optimize the transmit beamforming at the base station (BS) and the end-to-end transmission matrix of the SIM, aiming to minimize the Cramér-Rao Bound (CRB) for target estimation. The algorithm employs alternating optimization and semidefinite relaxation (SDR) techniques to solve the non-convex SINR-constrained CRB minimization problem. An experimental platform for SIM is designed and built, and the performance of the proposed algorithms for communication and sensing tasks is evaluated. The system model involves a BS with multiple antennas, multiple communication users (CUs), and an extended target in the non-line-of-sight (NLoS) region. The SIM consists of multiple transmissive layers, each with a set of meta-atoms. The end-to-end transmission matrix of the SIM is derived from the transmission coefficients of each layer. The system is designed to optimize the transmission beamforming and SIM transmission matrix to minimize the CRB for target estimation while meeting the SINR and power constraints. The paper also presents numerical and experimental results showing that increasing the number of layers in the SIM improves the performance of the ISAC system. The experimental results demonstrate that the proposed algorithm achieves improved communication and sensing performance, with the number of layers and inter-layer spacing affecting the signal quality. The results indicate that the proposed SIM-based ISAC system can effectively enhance both communication and sensing capabilities.This paper proposes a stacked intelligent metasurface (SIM)-assisted integrated sensing and communications (ISAC) system, where the SIM is used to enhance both communication and sensing performance. The system aims to estimate the complete target response matrix relative to the SIM while satisfying minimum signal-to-interference-plus-noise ratio (SINR) constraints for communication users (CUs) and maximum transmit power. The paper introduces a multi-layer alternating optimization (MAO) algorithm to jointly optimize the transmit beamforming at the base station (BS) and the end-to-end transmission matrix of the SIM, aiming to minimize the Cramér-Rao Bound (CRB) for target estimation. The algorithm employs alternating optimization and semidefinite relaxation (SDR) techniques to solve the non-convex SINR-constrained CRB minimization problem. An experimental platform for SIM is designed and built, and the performance of the proposed algorithms for communication and sensing tasks is evaluated. The system model involves a BS with multiple antennas, multiple communication users (CUs), and an extended target in the non-line-of-sight (NLoS) region. The SIM consists of multiple transmissive layers, each with a set of meta-atoms. The end-to-end transmission matrix of the SIM is derived from the transmission coefficients of each layer. The system is designed to optimize the transmission beamforming and SIM transmission matrix to minimize the CRB for target estimation while meeting the SINR and power constraints. The paper also presents numerical and experimental results showing that increasing the number of layers in the SIM improves the performance of the ISAC system. The experimental results demonstrate that the proposed algorithm achieves improved communication and sensing performance, with the number of layers and inter-layer spacing affecting the signal quality. The results indicate that the proposed SIM-based ISAC system can effectively enhance both communication and sensing capabilities.