In this paper, we introduce the concepts of derivation of degree $n$, generalized derivation of degree $n$ and ternary derivation of degree $n$, where $n$ is a positive integer, and then we study algebraic properties of these mappings. For instance, we study the image of derivations of degree $n$ on algebras and in this regard we prove that, under certain conditions, every derivation of degree $n$ on an algebra maps the algebra into its Jacobson radical. Also, we present some characterizations of these mappings on algebras. For example, under certain assumptions, we show that if $f$ is an additive generalized derivation of degree $n$ with an associated mapping $d$, then either $f$ is a linear generalized derivation with the associated linear derivation $d$ or $f$ and $d$ are identically zero. Some other related results are also established.
