The superconducting tunneling effect in heterostructures, describing the process where single electrons or Cooper pairs tunnel through the barrier, can always play a significant role in understanding the phase coherence and pairing mechanisms in superconductors. Taking advantage of the easy cleavage to atomically-thin monolayer structure of layered superconductors and resulting quantum confinement of electrons or Cooper pairs at two-dimensional limit, van der Waals superconducting materials hosting superconducting order in monolayers or heterostructures can exhibit extensive emergent phenomena associated with quantum phase transitions of vortex and anti-vortex pairs. Examples of superconducting tunnel junctions (STJs) based on layered superconductors have been demonstrated to achieve novel phenomena, including Andreev bound states, Majorana bound states and 0/π-phase junctions. Since the characteristic parameters of quasiparticle tunneling through the barrier are directly associated with the energy gap values of superconductors, such critical parameter can be obtained within the STJ device geometry, which helps us understand and control the pairing states and emerging phenomena in superconductors. In this review, from the perspective of STJs with single electron tunneling and Cooper pair tunneling, we discuss Andreev reflection, Majorana bound states, photon-induced tunneling effects, non-reciprocal transport and superconducting diode phenomena, as well as prospects for layered-superconductor-based STJs.