**Introduction**

Magnetoelectrics [1], i.e. insulating materials that display coupled electric and magnetic degrees of freedom, have attracted widespread interest in recent years because of their fascinating physics and potential applications to novel electronic devices. An intense research effort is currently devoted to (1) identifying new candidate materials or artiﬁcial structures displaying magnetoelectric effects and (2) using external perturbations (strains, electric ﬁelds) to enhance the properties of the existing compounds. In spite of many exciting advances, a material displaying strong magnetoelectric behavior at room temperature has not emerged yet.

Theoretical modeling is expected to play a key role in identifying novel systems displaying magnetoelectric and/or multiferroic properties, and in describing the mutual coupling between their mechanical, electrical and magnetic degrees of freedom. In this sense, recently proposed methods to treat ﬁnite ﬁelds, ﬁnite strains and orbital magnetism within density-functional theory look very promising. Yet, as none of these tools have found widespread use to date, we have the perception that there is a lack of consensus (and awareness) on how their potential could be exploited to develop a more consistent theory, and/or to test the usual assumptions of the present models. Furthermore, there is a clear need of translating these basic concepts into higher-level descriptions (Landau-type and/or effective-Hamiltonian theories).

We thus believe it is very timely to bring together researchers with different backgrounds (ﬁrst-principles theory, effective models, Landau theory) to critically discuss the state of the art of this ﬁeld and pave the way towards a general, consistent and useful theory of magnetoelectricity. In parallel, we would like to foster a debate on what are the most promising avenues for the future design of multiferroic materials. Members of the experimental community will also be invited to give us their views on the ﬁeld and on how ﬁrst-principles theory could best contribute.

**State of the art**

Recent developments in the first-principles theory of magnetoelectric multiferroics have taken place on three different fronts.

First, increasingly accurate methods have been devised to compute the response of a given material or nanostructure to a well-defined external perturbation (electric field, pressure/tension, magnetic field). Central to these advances was the development of the modern theory of polarization [2,3], and its coupling to an external field [4,5]. Wu, Vanderbilt and Hamann [6] later used these ingredients to systematically treat electric fields and strains within a perturbative framework. Iniguez et al. very recently generalized these ideas to the calculation of ion-mediated [7] and strain-mediated [8] magnetoelectric coefficients; yet, the calculations have so far neglected the purely electronic and orbital magnetization contributions, and have been limited to the linear-response regime. Recent progress in the treatment of electric, structural and magnetic degrees of freedom holds promise for overcoming these limitations.

For example: i) Stengel, Spaldin and Vanderbilt [9] simplified the proper [10] treatment of linear and non-linear piezoelectric effects by expressing the electrical variables in reduced lattice coordinates; ii) there is growing evidence that the metric tensor might have several advantages over the traditional Cauchy strain, both as dynamical variable [11] and as foundation for a perturbative analysis [12]; iii) the description of orbital magnetism has reached a relatively mature level of development [13]. None of the above methodological tools i)-iii) was applied to magnetoelectric systems to date, and a comprehensive and internally consistent theory that embodies all these advances has still to be written. It is important to mention here that the electronic ground state of many magnetic insulators is incorrectly described by common density functionals, and that significant effort has also been devoted to develop efficient simulation methods that address this long-standing issue (for example, see Ref. [14]).

Next, a significant effort was devoted to building higher-level effective models with parameters calculated from first-principles. A few representative developments concern the works of Bellaiche et al. [15, 16] on BiFeO3, Fennie and Rabe [17] on EuTiO3, or Harris et al. on magnetically-driven ferroelectrics like Ni3V2O8 [18]. Also important in multiferroics are the couplings between non-polar structural instabilities (e.g. rotations of oxygen octahedra) and electrical and/or magnetic degrees of freedom, which were investigated, e.g., by Fennie and Rabe [19] for YMnO3. No attempts have been made to date at adopting a reduced lattice representation (see above paragraph) for the relevant degrees of freedom in effective Hamiltonians. Also, we think a general atomistic treatment of spin cycloids and their coupling to polarization and strain is still missing.

Finally, some remarkable work has been done to identify useful criteria for designing novel materials and/or nanostructures with enhanced magnetoelectric couplings. Without the pretension to be exhaustive, a few recent examples include: Nenert and Palstra [20] using symmetry and empirical arguments; Fennie [21] based on spin-orbit effects; Delaney et al. [22] based on exchange-striction; Picozzi et al. [23] based on electronic effects; or Bhattacharjee et al. [24], who showed examples of ferroelectricity driven by magnetic species. The value of these and other contributions is unquestionable; yet, it is not clear to date what the most promosing route towards room-temperature magnetoelectricity is.