Let μ be a measure on a domain Ω in ℂ n such that the Bergman space of holomorphic functions in L 2 ( Ω , μ ) possesses a reproducing kernel K ( x , y ) and K ( x , x ) > 0 ∀ x ∈ Ω . The Berezin transform associated to μ is the integral operator B f ( y ) = K ( y , y ) - 1 ∫ Ω f ( x ) | K ( x , y ) | 2 d μ ( x ) . The number B f ( y ) can be interpreted as a certain mean value of f around y , and functions satisfying B f = f as functions having a certain mean-value property. In this paper we investigate the boundary behavior of B f , the existence of functions f satisfying B f = f and having prescribed boundary values, and the convergence of the iterates B k f , k → ∞ . The best results are obtained for smoothly bounded strictly pseudoconvex domains Ω with any measure μ as above, and for bounded symmetric domains Ω and μ one of the standard rotation-invariant measures on them. We also carry out similar investigation for convolution operators B μ f = f * μ on a bounded symmetric domain Ω = G / K with a K -invariant absolutely continuous probability measure μ , and study the behavior of the geodesic symmetries φ a of Ω as a tends to the boundary.