Add to ORKGAbstract. Lavrentiev regularization is a popular approach to the solution of linear discrete ill-posed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods. Key words. Discrete ill-posed problem, matrix nearness problem, Lavrentiev regularization, fractional Lavrentiev regularizatio