The relation between $\mathbb{Q}$-curves and certain
abelian varieties of $\operatorname{GL}_2$-type
was established by Ribet (`Abelian varieties over
$\mathbb{Q}$ and modular forms', {\em Proceedings of
the KAIST Mathematics Workshop} (1992) 53--79)
and generalized to building blocks, the
higher-dimensional analogues of $\mathbb{Q}$-curves,
by Pyle in her PhD Thesis (University of California
at Berkeley, 1995).
In this paper we investigate some aspects
of $\mathbb{Q}$-curves with no complex multiplication
and the corresponding abelian varieties of
$\operatorname{GL}_2$-type, for which we mainly use
the ideas and techniques introduced by Ribet
(op. cit. and `Fields of definition of abelian varieties
with real multiplication', {\em Contemp.\ Math.} 174
(1994) 107--118).
After the Introduction,
in Sections 2 and 3 we obtain a characterization of the
fields where a $\mathbb{Q}$-curve and all the isogenies
between its Galois conjugates can be defined up to
isogeny, and we apply it to certain fields of type
$(2,\dots,2)$.
In Section 4 we determine the endomorphism algebras
of all the abelian varieties of
$\operatorname{GL}_2$-type having as a quotient
a given $\mathbb{Q}$-curve in easily computable terms.
Section 5 is devoted to a particular case of
Weil's restriction of scalars functor applied to
a $\mathbb{Q}$-curve,
in which the resulting abelian variety factors
over $\mathbb{Q}$ up to isogeny as a product of abelian
varieties of $\operatorname{GL}_2$-type.
Finally, Section 6 contains examples:
we parametrize the $\mathbb{Q}$-curves coming from
rational points of the modular curves $X^*(N)$
having genus zero, and we apply the results of
Sections 2--5 to some of the curves obtained.
We also give results concerning the existence
of quadratic $\mathbb{Q}$-curves.
1991 Mathematics Subject Classification:
primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.