Understanding how young stars gain their masses through disk-to-star accretion is of paramount importance in astrophysics. It affects our knowledge about the early stellar evolution, the disk lifetime and dissipation processes, the way the planets form on the smallest scales, or the connection to macroscopic parameters characterizing star-forming regions on the largest ones, among others. In turn, mass accretion rate estimates depend on the accretion paradigm assumed. For low-mass T Tauri stars with strong magnetic fields there is consensus that magnetospheric accretion (MA) is the driving mechanism, but the transfer of mass in massive young stellar objects with weak or negligible magnetic fields probably occurs directly from the disk to the star through a hot boundary layer (BL). The intermediate-mass Herbig Ae/Be (HAeBe) stars bridge the gap between both previous regimes and are still optically visible during the pre-main sequence phase, thus constituting a unique opportunity to test a possible change of accretion mode from MA to BL. This review deals with our estimates of accretion rates in HAeBes, critically discussing the different accretion paradigms. It shows that although mounting evidence supports that MA may extend to late-type HAes but not to early-type HBes, there is not yet a consensus on the validity of this scenario versus the BL one. Based on MA and BL shock modeling, it is argued that the ultraviolet regime could significantly contribute in the future to discriminating between these competing accretion scenarios.