  We construct solutions for thin black rings in Anti-deSitter and deSitter
spacetimes using approximate methods. Black rings in AdS exist with arbitrarily
large radius and satisfy a bound |J| \leq LM, which they saturate as their
radius becomes infinitely large. For angular momentum near the maximum, they
have larger area than rotating AdS black holes. Thin black rings also exist in
deSitter space, with rotation velocities varying between zero and a maximum,
and with a radius that is always strictly below the Hubble radius. Our general
analysis allows us to include black Saturns as well, which we discuss briefly.
We present a simple physical argument why supersymmetric AdS black rings must
not be expected: they do not possess the necessary pressure to balance the AdS
potential. We discuss the possible existence or absence of `large AdS black
rings' and their implications for a dual hydrodynamic description. An analysis
of the physical properties of rotating AdS black holes is also included.
