In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group
G
G
lying in a variety
M
{\mathcal M}
can be embedded in a
4
4
-generated group
H
∈
M
A
H \in {\mathcal M}{\mathcal A}
(
A
{\mathcal A}
means the variety of abelian groups). If
G
G
is a finite group, then
H
H
can also be found as a finite group. It follows, that any finitely generated (finite) solvable group
G
G
of the derived length
l
l
can be embedded in a
4
4
-generated (finite) solvable group
H
H
of length
l
+
1
l+1
. Thus, we answer the question of V. H. Mikaelian and A. Yu. Olshanskii. It is also shown that any countable group
G
∈
M
G\in {\mathcal M}
, such that the abelianization
G
a
b
G_{ab}
is a free abelian group, is embeddable in a
2
2
-generated group
H
∈
M
A
H\in {\mathcal M}{\mathcal A}
.