The recent achievements gained in understanding of the dissolution dynamics of miscible interfaces are reviewed. Our consideration is restricted to isothermal systems with the mass transfer purely driven by inhomogeneities in the field of concentration. Both experimental and theoretical works are examined. The attention is given to the effects of dynamic surface tension, interfacial diffusion, dynamics of the contact line, and to solutal convective flows. We conclude that, despite ubiquitousness and importance of physical processes involving miscible interfaces, the physics that defines the thermo- and hydrodynamic evolution of such interfaces is still not properly understood, especially in respect to dissolution rate at a miscible liquid/liquid phase boundary and to wetting properties at liquid/solid boundary. A consistent theoretical description for the slowly miscible binary systems is given within the phase-field (Cahn-Hilliard) approach. Nevertheless, there are just a few modelling works that take into account all the effects pertinent to miscible liquid/liquid interfaces.