  We look at algebraic embeddings of the Cremona group in $n$ variables
$Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic
variety $M$. First we study geometrical properties of an example of an
embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second
part, we give a full classification of all algebraic embeddings of $Cr_2(C)$
into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to
algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for
arbitrary $n\geq 2$. In particular, this yields a classification of all
algebraic $PGL_{n+1}(C)$-actions on smooth projective varieties of dimension
$n+1$ that can be extended to rational actions of $Cr_n(C)$.
