The first-order transitions at T-1=40 degrees K and T-2=87 degrees K in RbFeF3 have been measured as a function of hydrostatic pressure and applied magnetic field. It was not possible to observe the T-N= 102 degrees K transition with a magnetic-susceptibility measurement. It was found that (Delta T-1/Delta H-a)(p) = 0.35 degrees/kOe, (Delta T-2/Delta H-a)(p) = 0.19 degrees/kOe, (Delta T-1/Delta P)(H) = 0.18 degrees/kbar and (Delta T-2/Delta P)(H) = - 0.81 degrees/kbar. These results correspond to latent heats of 0.006 and 0.04 cal/g at T-1 and T-2, respectively, and relative volume changes Delta V-1/V-1 = 1.5X10(-6), Delta V-2/V-2=-22X10(-6). It is pointed out that a Jahn-Teller distortion to tetragonal (c/a>1) symmetry in the interval T-2 < T < T-N introduces a strong magnetoelastic coupling. This causes the heavy twinning that has been observed below T-N, and the resulting twinned structure is retained in the entire temperature interval 0< T< T-N. In the temperature interval T-1<T<T-2, Rb+-F- interactions induce distortions to orthorhombic or tetragonal symmetries that are superimposed on the Jahn-Teller distortion. The orthorhombic distortion is cooperative across twin boundaries caused by the Jahn-Teller distortion and also permits spin canting, which introduces a ferromagnetic component below T-2. It is shown how the interplay of these distortions plus strong magnetoelastic coupling can explain the appearance of two sets of Mossbauer peaks below T-2 and results in macroscopic ferromagnetic components having cubic symmetry even though the microscopic crystallographic symmetry is "orthorhombic" (T-1<T<T-2). The Jahn-Teller distortion changes to rhombohedral (alpha<60 degrees) for T<T-1; in combination with the existing orthorhombic structure, this produces monoclinic symmetry on a microscopic scale. Nevertheless, it is shown that the macroscopic magnetization retains its cubic symmetry, that the easy magnetization direction changes from (100 to the (110), that the apparent moment increases, and that there may still be two sets of Mossbauer peaks.