  The properties of static spherically symmetric black holes, which are both
electrically and magnetically charged, and which are coupled to the dilaton in
the presence of a cosmological constant, Lambda, are considered. It is shown
that apart from the Reissner-Nordstrom-de Sitter solution with constant
dilaton, such solutions do not exist if Lambda > 0 (in arbitrary spacetime
dimension >=4 ). However, asymptotically anti-de Sitter dyonic black hole
solutions with a non-trivial dilaton do exist if Lambda < 0. Both these
solutions and the asymptotically flat (Lambda = 0) solutions are studied
numerically for arbitrary values of the dilaton coupling parameter, g_0, in
four dimensions. The asymptotically flat solutions are found to exhibit two
horizons if g_0 = 0, 1, \sqrt{3}, \sqrt{6}, ..., \sqrt{n(n+1)/2},..., and one
horizon otherwise. For asymptotically anti-de Sitter solutions the result is
similar, but the corresponding values of g_0 are altered in a non-linear
fashion which depends on Lambda and the mass and charges of the black holes.
All dyonic solutions with Lambda <= 0 are found to have zero Hawking
temperature in the extreme limit, however, regardless of the value of g_0.
