Suppose D is an open connected subset of C. The classical (strong) maximum modulus theorem says that the modulus of an analytic function f : D → C can not assume its maximum at a point in D unless f is constant on D. This does not generalize to analytic Banach space valued functions. However a (weaker) version holds for a Banach space X valued analytic function, which says that if f : D → X is analytic, then ||f(z)|| has no maximum on D unless ||f(z)|| is identically constant on D. In fact the strong form of the maximum modulus principle holds for X if and only if each point in X of norm one is a ‘complex extreme point’ of the unit sphere of X. We shall discuss these results with some examples. The talk will be elementary and accessible to all.