  An element $\alpha \in \mathbb {F}_{q^n}$ is normal over $\mathbb {F}_q$ if
$\alpha$ and its conjugates $\alpha, \alpha^q, \cdots \alpha^{q^{n-1}}$ form a
basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$. Recently, Huczynska, Mullen,
Panario and Thomson (2013) introduce the concept of $k$-normal elements,
generalizing the normal elements. In the past few years, many questions
concerning the existence and number of $k$-normal elements with specified
properties have been proposed. In this paper, we discuss some of these
questions and, in particular, we provide many general results on the existence
of $k$-normal elements with additional properties like being primitive or
having large multiplicative order. We also discuss the existence and
construction of $k$-normal elements in finite fields, providing a connection
between $k$-normal elements and the factorization of $x^n-1$ over $\mathbb
{F}_q$.
