This paper proposes a channel estimation and beamforming design scheme for beyond diagonal reconfigurable intelligent surfaces (BD-RIS) aided communication systems. BD-RIS extends conventional RIS by introducing inter-element connections, enabling more flexible wave manipulation and larger coverage. However, effective channel estimation for BD-RIS remains an open problem. The authors propose a least squares (LS) based channel estimation method, which theoretically achieves the minimum mean square error (MSE). They also design a joint pilot sequence and BD-RIS configuration to minimize MSE. The estimated channel is then used for beamforming design in two BD-RIS scenarios. Simulation results show that the proposed method achieves the theoretical minimum MSE and provides satisfactory performance close to perfect channel state information (CSI) cases. The study also highlights the trade-off between channel estimation performance and data transmission performance, showing that increased inter-element connections in BD-RIS improve performance but increase training overhead. The proposed method is applicable to various BD-RIS architectures, including reflective, hybrid, and multi-sector modes. The results demonstrate that the proposed channel estimation and beamforming design schemes are effective for BD-RIS aided communication systems.This paper proposes a channel estimation and beamforming design scheme for beyond diagonal reconfigurable intelligent surfaces (BD-RIS) aided communication systems. BD-RIS extends conventional RIS by introducing inter-element connections, enabling more flexible wave manipulation and larger coverage. However, effective channel estimation for BD-RIS remains an open problem. The authors propose a least squares (LS) based channel estimation method, which theoretically achieves the minimum mean square error (MSE). They also design a joint pilot sequence and BD-RIS configuration to minimize MSE. The estimated channel is then used for beamforming design in two BD-RIS scenarios. Simulation results show that the proposed method achieves the theoretical minimum MSE and provides satisfactory performance close to perfect channel state information (CSI) cases. The study also highlights the trade-off between channel estimation performance and data transmission performance, showing that increased inter-element connections in BD-RIS improve performance but increase training overhead. The proposed method is applicable to various BD-RIS architectures, including reflective, hybrid, and multi-sector modes. The results demonstrate that the proposed channel estimation and beamforming design schemes are effective for BD-RIS aided communication systems.