In mathematics, inequalities have always being a powerful tool. It remains a major mathematical physics research area beginning with Gauss, Cauchy and oth- ers. Matrix inequalities appears in various ¿elds such as system and control theory [30], optimization [98], mathematical physics [18], statistics [84], economics [95] and engineering disciplines [36]. A wide variety of problems arising in system and con- trol theory can be reduced to a few standard convex or quasi-convex optimization problems involving linear matrix inequalities. Matrix inequalities also play a vital role in quantum statistical inference and quantum information theory (see [17, 78] and references within). As may be deduced from the title, "Matrix Inequalities", various topics such as matrix inequalities of inde¿nite form, inequalities involving matrix means, Kadison type inequality via matrix means and generalised Furuta's inequality are presented in the thesis.