  We consider global monopoles in asymptotic de Sitter/ Anti- de Sitter
space-time. We present the by our numerical analysis confirmed asymptotic
behaviour of the metric and Goldstone field functions. We find that the
appearance of horizons in this model depends strongly on the sign and value of
the cosmological constant as well as on the value of the gravitational
coupling. In Anti-de Sitter (AdS) space, we find that for a fixed value of the
cosmological constant, global monopoles without horizons exist only up to a
critical value of the gravitational coupling. Moreover, we observe (in contrast
to another recent study) that the introduction of a cosmological constant can
NOT render a positive mass of the global monopole.
