In this chapter, we consider a new class of quasilinear elliptic problems, namely elliptic equations that involve a differential operator having partial derivatives with different variable exponents. In this case, the underlying functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. We are concerned with the qualitative analysis of solutions and we establish several existence and nonexistence results. These properties are influenced by the competition between the growth rates of the anisotropic coefficients. Some of the results in this chapter extend original contributions in [85, 87, 88, 62, 63, 91, 9, 8]. We also refer to the book by Repovsˇ and Semenov [92] for some material related to this subject.