This talk describes some joint work with Yves F{\'e}lix, which establishes several fundamental results about the rational homotopy behaviour of the so-called evaluation map. Let $map(X, X;1)$ denote the path component of the space of self-maps of a space $X$ that contains the identity map. Then the evaluation map of $X$ is the map $w: map(X, X;1)--> X$ defined by evaluation of a self-map $f$ at the basepoint $x_0$ of $X$, that is, $w(f) = f(x_0)$. It turns out that this evaluation map is important in a number of ways. For example, it may be considered a ``universal connecting map" for fibrations with fibre $X$. More generally, let $E$ be an $H$-space acting on a based space $X$. Then we refer to $ev \colon E \to X$, the map obtained by acting on the base point of $X$, as a ``generalized evaluation map." This notion includes, for example, the orbit map $G \to X$ of certain group actions of a connected group $G$ on $X$. We prove some results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map or to an orbit map. With mild hypotheses on $X$, we show that a generalized evaluation map factors, up to rational homotopy, through a (relatively small) finite product of odd-dimensional spheres. This result has strong consequences: if the image in rational homotopy groups of $ev$ is trivial, then \emph{the generalized evaluation map is null-homotopic} after rationalization; unless $X$ satisfies a very strong splitting condition, any generalized evaluation map \emph{induces the trivial homomorphism in rational cohomology}. We will include illustrative examples and mention several subsidiary results of interest.