  We study horospherical Radon transforms that integrate functions on the
$n$-dimensional real hyperbolic space over horospheres of arbitrary fixed
dimension $1\le d\le n-1$. Exact existence conditions and new explicit
inversion formulas are obtained for these transforms acting on smooth functions
and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known
Gelfand-Graev transform.
