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The book is devoted to the review of original results of the author in the theory of integrable s=1/2 quantum spin chains with the exchange constants proportional to inverse square hyperbolic sine (infinite chains) and Weierstrass elliptic function with real period which equals to the number of lattice sites (chains with periodic boundary conditions). It contains the proof of integrability (explicit construction of integrals of motion) and detailed description of multimagnon wave functions generalizing the well-known Bethe ansatz for one-dimensional chains with nearest-neighbor exchange. Much attention is paid to finding explicit form of corresponding eigenvalue equations for arbitrary number of magnons based on analogy with quantum Calogero-Moser integrable particle systems with inverse square hyperbolic and elliptic potentials at some integer coupling constant.