This thesis is concerned with the formulation of a suitable notion of monotonicity of discrete and continuous-time dynamical systems on Lie groups and homogeneous spaces. In a linear space, monotonicity refers to the property of a system that preserves an ordering of the elements of the space. Monotone systems have been studied in detail and are of great interest for their numerous applications, as well as their close connections to many physical and biological systems. In a linear space, a powerful local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear systems. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to seek a suitable adaptation of monotonicity on such spaces. However, the question of how one can develop a suitable notion of monotonicity on a nonlinear manifold is complicated by the general absence of a clear and well-defined notion of order on such a space. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce notions of conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces and develop the theory in this thesis. We illustrate the ideas with numerous examples and apply the theory to a number of areas, including the theory of consensus on Lie groups and order theory on the set of positive definite matrices.