  The purpose of this paper is to introduce $N_k(\ell)$-maps ($1\le
k,\ell\le\infty$), which describe higher homotopy normalities, and to study
their basic properties and examples. An $N_k(\ell)$-map is defined with higher
homotopical conditions. It is shown that a homomorphism is an $N_k(\ell)$-map
if and only if there exists fiberwise maps between fiberwise projective spaces
with some properties. Also, the homotopy quotient of an $N_k(k)$-map is shown
to be an $H$-space if its LS category is not greater than $k$. As an
application, we investigate when the inclusions
$\operatorname{SU}(m)\to\operatorname{SU}(n)$ and
$\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$ are $p$-locally
$N_k(\ell)$-maps.
