<p>In this survey paper, we present some recent results on weighted spaces of analytic functions with mixed norm, defined in terms of the Fourier (Mellin) transform in three cases (settings). These three cases are characterized by three types of hyperbolic geometry in the unit disc and the coordinate systems associated with them—elliptic, parabolic, and hyperbolic. The first is described up to an automorphism of the disc by polar coordinates in the unit disc, and the second and third—after a conformal mapping onto the half-plane (for ease of perception)—are transformed into Cartesian and polar coordinates on the half-plane, respectively. Thus, we consider spaces on the unit disc with a mixed norm associated with polar coordinates, and on the upper half-plane with a mixed norm associated with Cartesian and also with polar coordinates. We also present results on the boundedness of Toeplitz operators with special symbols also associated with this geometry (coordinate systems). In other words, we consider radial symbols in the disc and symbols depending on a vertical or angular variable on the half-plane. We also discuss the connection with earlier studies in the classical case of weighted spaces of Bergman–Jerbashyan type and the study of Toeplitz operators in these spaces with the symbols as specified above.</p>