  Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study
flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the
hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with
speed given by a function $W$ of their principal curvatures,called a Weingarten
function, which is homogeneous, monotonic increasing with respect to any of its
variables, and positive on the positive cone. We obtain existence results for
flows with isoparametric initial data, in which the hypersurfaces
$F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either
a simply connected space form or a rank-one symmetric space of noncompact type.
We prove that the avoidance principle holds for Weingarten flows defined by odd
Weingarten functions, and also that such flows are embedding preserving.
