The paper introduces the joint graphical lasso (JGL), a method for estimating multiple related Gaussian graphical models from high-dimensional data with observations in distinct classes. JGL borrows strength across classes to estimate precision matrices that share characteristics such as edge locations or weights. The method maximizes a penalized log likelihood using generalized fused lasso or group lasso penalties, and solves the resulting convex optimization problem with a fast ADMM algorithm. The performance of JGL is demonstrated through simulated and real data examples. The paper begins by discussing the problem of estimating precision matrices in Gaussian graphical models, highlighting the challenges of high-dimensional data and the need for sparsity. It then introduces the graphical lasso, which uses an l1 penalty to estimate precision matrices with sparse structures. The paper extends this to the joint graphical lasso, which estimates multiple precision matrices simultaneously, encouraging shared characteristics across classes. The joint graphical lasso is formulated as a penalized log likelihood problem with a convex penalty function. Two specific penalty functions are considered: the fused graphical lasso (FGL), which encourages similarity between precision matrices, and the group graphical lasso (GGL), which encourages shared sparsity patterns. The paper presents an ADMM algorithm for solving the joint graphical lasso problem, which is efficient and converges to the global optimum. Theoretical results show that when tuning parameters are large, the JGL optimization problem can be solved without computing the eigen decomposition of a p×p matrix, leading to significant computational improvements. The paper also discusses tuning parameter selection using the Akaike Information Criterion (AIC) and evaluates the performance of JGL in simulation studies and real data analysis. In a simulation study, JGL outperforms existing methods in terms of edge detection, sparsity, and network similarity. In a real data analysis, JGL is applied to a lung cancer microarray dataset, revealing shared and class-specific edges, and identifying potential gene interactions that may warrant further investigation. The results demonstrate the effectiveness of JGL in estimating multiple related graphical models from high-dimensional data.The paper introduces the joint graphical lasso (JGL), a method for estimating multiple related Gaussian graphical models from high-dimensional data with observations in distinct classes. JGL borrows strength across classes to estimate precision matrices that share characteristics such as edge locations or weights. The method maximizes a penalized log likelihood using generalized fused lasso or group lasso penalties, and solves the resulting convex optimization problem with a fast ADMM algorithm. The performance of JGL is demonstrated through simulated and real data examples. The paper begins by discussing the problem of estimating precision matrices in Gaussian graphical models, highlighting the challenges of high-dimensional data and the need for sparsity. It then introduces the graphical lasso, which uses an l1 penalty to estimate precision matrices with sparse structures. The paper extends this to the joint graphical lasso, which estimates multiple precision matrices simultaneously, encouraging shared characteristics across classes. The joint graphical lasso is formulated as a penalized log likelihood problem with a convex penalty function. Two specific penalty functions are considered: the fused graphical lasso (FGL), which encourages similarity between precision matrices, and the group graphical lasso (GGL), which encourages shared sparsity patterns. The paper presents an ADMM algorithm for solving the joint graphical lasso problem, which is efficient and converges to the global optimum. Theoretical results show that when tuning parameters are large, the JGL optimization problem can be solved without computing the eigen decomposition of a p×p matrix, leading to significant computational improvements. The paper also discusses tuning parameter selection using the Akaike Information Criterion (AIC) and evaluates the performance of JGL in simulation studies and real data analysis. In a simulation study, JGL outperforms existing methods in terms of edge detection, sparsity, and network similarity. In a real data analysis, JGL is applied to a lung cancer microarray dataset, revealing shared and class-specific edges, and identifying potential gene interactions that may warrant further investigation. The results demonstrate the effectiveness of JGL in estimating multiple related graphical models from high-dimensional data.