Given a set Ω in Rd, we say that Ω is spectral if L2(Ω) has an orthonormal Fourier basis consisting of exponentials. Fuglede's conjecture says that a set Ω is spectral if and only if Ω tiles Rd by translations. This implies that the collection {Ω+γ}, for each γ in a set Γ, forms a partition of Rd, except for a set of measure zero where overlaps might occur. The conjecture is valid if Γ is a lattice, known as Fuglede's Theorem. However, this conjecture has been generally disproved in both directions for dimensions three and higher. For dimensions d=1 and d=2, only partial results are available. Specifically, in the case of d=1, Łaba demonstrated that if Ω is the union of two intervals, the conjecture holds. A similar conjecture exists in the setting of Gabor basis. Concretely, the conjecture in the setting of Gabor analysis says that the characteristic function of a set Ω in Rd can be used as a window of an orthogonal Gabor basis if and only if Ω is spectral and tiles the space. In this talk, I will recall some basic results related to the Fuglede conjecture, and then I will discuss some tools and results related to the Gabor-Fuglede version. In particular, I will comment on some results I have obtained in collaboration with Elona Agora and Mihalis Kolountzakis on a version of Łaba’s result in the setting of Gabor analysis.