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.