Quasiparticle bound states are found theoretically on transparent interfaces of d-wave superconductors (dSC) with charge density wave solids (CDW), as well as s-wave superconductors (sSC) with d-density waves (DDW). These bound states represent a combined effect of Andreev reflection from the superconducting side and an unconventional quasiparticle Q-reflection from the density wave solid. If the order parameter for a density wave state is much less than the Fermi energy, bound states with almost zero energy take place for an arbitrary orientation of symmetric interfaces. For larger values of the order parameter, dispersionless zero-energy states are found only on (110) interfaces. Two dispersive energy branches of subgap quasiparticle states are obtained for (100) symmetric interfaces. Andreev low-energy bound states, taking place in junctions with CDW or DDW interlayers, result in anomalous junction properties, in particular, the low-temperature behavior of the Josephson critical current. PACS numbers: 74.45.+c, 74.50.+r Introduction. Low-energy quasiparticle states play an important role in forming electron transport in mesoscopic hybrid superconducting systems at low temperatures. In transparent superconductor-normal metalsuperconductor (S-N-S) junctions, subgap states originate entirely in Andreev reflection processes. In the presence of finite interface transparencies, both Andreev and conventional reflections come into play in forming subgap bound states. Zero-energy Andreev surface states in d-wave superconductors also represent a combined effect of Andreev and specular quasiparticle reflections.