Vesicle membranes with two coexisting fluid phases can exhibit a variety of domain patterns. These patterns and the associated vesicle shapes are studied theoretically by minimization of energy functionals that depend on the membrane composition and on the material parameters of the membrane. The latter parameters are (i) the bending rigidities and (ii) the Gaussian curvature moduli of the two types of membrane domains as well as (iii) the line tension of the domain boundaries. It is shown that the interplay between these different parameters leads to stable multi-domain patterns with more than two domains for a wide range of membrane composition and material parameters. As the membrane composition or the material parameters are varied, the vesicles can undergo transitions between different patterns of membrane domains. For fixed vesicle volume, an additional domain-induced transition is observed, in which the vesicle shape changes even though the domain pattern does not. The different multi-domain patterns and vesicle shapes are summarized in terms of morphology diagrams.