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The thesis studies the integer-power moments of the central values of families of modular L-functions. The two families under consideration in the thesis are those quadratic twists of a L-function associated with a cusp from and L-functions of a Hecke-basis of the space of cusp forms. Applications of the moment estimates derived in the thesis include (1) a non-vanishing result (2) a zero density estimate for quadratic twisted L-functions.