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 fields 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 indefinite form, inequalities involving matrix means, Kadison type inequality via matrix means and generalised Furuta's inequality are presented in the thesis.