Mutational Analysis : A Joint Framework for Cauchy Problems in and Beyond Vector Spaces Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications however it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions for example do not have an obvious linear structure. This book generalizes ordinary differential equations beyond the borders of vector spaces with a focus on the well posed Cauchy problem in finite time intervals. Here are some of the examples: Feedback evolutions of compact subsets of the Euclidean space Birth and growth processes of random sets (not necessarily convex) Semilinear evolution equations Nonlocal parabolic differential equations Nonlinear transport equations for Radon measures A structured population model Stochastic differential equations with nonlocal sample dependence and how they can be coupled in systems immediately due to the joint framework of Mutational Analysis. Finally the book offers new tools for modelling Extending Ordinary Differential Equations to Metric Spaces: Aubin’s Suggestion Adapting Mutational Equations to Examples in Vector Spaces: Local Parameters of Continuity Less Restrictive Conditions on Distance Functions: Continuity Instead of Triangle Inequality Introducing Distribution Like Solutions to Mutational Equations Mutational Inclusions in Metric Spaces