We investigate the anisotropic superconducting and magnetic properties of single-crystal RbEuFe4As4 using magnetotransport and magnetization measurements. We determine a magnetic ordering temperature of the Eu moments of Tm=15K and a superconducting transition temperature of Tc=36.8K. The superconducting phase diagram is characterized by high upper critical field slopes of -70 and -42 kG/K for in-plane and out-of-plane fields, respectively, and a surprisingly low superconducting anisotropy of Γ=1.7. Ginzburg-Landau parameters of κc∌67 and κab∌108 indicate extreme type-II behavior. These superconducting properties are in line with those commonly seen in optimally doped Fe-based superconductors. In contrast, Eu magnetism is quasi-two dimensional (2D), as evidenced by highly anisotropic in-plane and out-of-plane exchange constants of 0.6 K and <0.04 K. A consequence of the quasi-2D nature of the Eu magnetism are strong magnetic fluctuation effects, a large suppression of the magnetic ordering temperature as compared to the Curie-Weiss temperature, and a kinklike anomaly in the specific heat devoid of any singularity. Magnetization curves reveal a clear magnetic easy-plane anisotropy with in-plane and out-of-plane saturation fields of 2 and 4 kG. © 2018 American Physical Society.