In this work, the consistent, predictive and empirically adequate microscopic theory of pseudogap phenomena and unconventional Bose-liquid superconductivity (superfluidity) is presented, based on the fact that in high-Tc cuprates and other related systems the energy εA of the effective attraction between fermionic quasiparticles is comparable with their Fermi energy εF and the bosonic Cooper pairs are formed above Tc (the temperature of the superfluid transition) and then a small part of such Cooper pairs condense into a Bose superfluid at Tc. According to this theory, the doped high-Tc cuprates and other systems with low Fermi energies (εF ∼ εA) are unconventional bosonic superconductors/superfluids and exhibit pseudogap phases above Tc, λ-like superconducting transition at Tc and Bose-liquid superconductivity below Tc. The relevant charge carriers in high-Tc cuprates are polarons which are bound into bosonic Cooper pairs above Tc. Polaronic effects and related pseudogap weaken with increasing the doping and disappear at a quantum critical point where a small Fermi surface of polarons transforms into a large Fermi surface of quasi-free carriers. The modified BCS-like theory describes another pseudogap regime but the superconducting/superfluid transition in high-Tc cuprates and related systems is neither the BCS-like transition nor the usual Bose-Einstein condensation. A good quantitative agreement is found between pseudogap theory and experiment. Universal criteria for bosonization of Cooper pairs are formulated in terms of two fundamental ratios εA/εF and ∆F /εF (where ∆F is the BCS-like gap). The mean-field theory of the coherent single particle and pair condensates of bosonic Cooper pairs describes fairly well the novel superconducting states (i.e., two distinct superconducting A and B phases below Tc and a vortexlike state above Tc) and various salient features (λ-like transition at Tc, kink-like anomalies in all superconducting/superfluid parameters near the first-order phase transition temperature T * c lower than Tc, gapless excitations below T * c and two-peak specific heat anomalies) of high-Tc cuprates in full agreement with the experimental findings. Though Bose-liquid superconductivity in the bulk of high-Tc cuprates is destroyed at Tc, but it can persist above Tc at grain boundaries and interfaces of these materials up to room temperature. The unusual superconducting/superfluid states and properties of other exotic systems (e.g., heavy-fermion and organic compounds, Sr2RuO4, 3 He, 4 He and atomic Fermi gases) are explained more clearly by the theory of Bose superfluids. Finally, the new criteria and principles of unconventional superconductivity and superfluidity are formulated.