  The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff
spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any
compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the
class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli,
then it is $\kappa$-Fr\'echet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is
Ascoli iff it is $\kappa$-Fr'echet-Urysohn. This leads to the following
extension of a result of Morishita: If for a \v{C}ech-complete space $X$ the
space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and
stratifiable, then $C_p(X)$ is an Ascoli space. Consequently: (a) If $X$ is a
complete metrizable space, then $C_p(X)$ is Ascoli iff $X$ is scattered. (b) If
$X$ is a \v{C}ech-complete Lindel\"of space, then $C_p(X)$ is Ascoli iff $X$ is
scattered iff $C_p(X)$ is Fr\'echet-Urysohn. Moreover, we prove that for a
paracompact space $X$ of point-countable type the following conditions are
equivalent: (i) $X$ is locally compact. (ii) $C_k(X)$ is a $k_{\mathbb
R}$-space. (iii) $C_k(X)$ is an Ascoli space. The Asoli spaces $C_k(X,[0,1])$
are also studied.
