We studied the initial value problem for dispersive flows of second, third and fourth orders from a point of view of geometric analysis and linear partial differential equations. We established the existence theorems for dispersive flows under the almost minimum restrictions on the geometric settings of the source and target manifolds. For example, we studied the second order equation which is called the Schroedinger map equation, whose solutions describe the flow from a closed Riemannian manifold to a compact almost Hermitian manifold, and succeeded in establishing the short-time existence theorem. This means that our geometric settings have no restriction as far as the description of the equation makes sense. In previous studies, the source manifold is supposed to be a circle (the one-dimensional torus) or an Euclidean space, and the target manifold is assumed to be a Kaehler manifold. For this reason, our results can be said to be big improvements.