  We prove global existence of Yamabe flows on non-compact manifolds $M$ of
dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is
conformally equivalent to a complete background metric $g_M$ of bounded,
non-positive scalar curvature and positive Yamabe invariant with conformal
factor $u_0$ bounded from above and below. We do not require initial curvature
bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from
above and below without growth condition.
