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This thesis will present an original work characterizing hyponormality and positivity of Toeplitz operators with bounded symbols on the weighted Bergman spaces of on the disk and whole complex plane.Cuckovic and Curto have recently obtained a necessary condition for the hyponormality of the Toeplitz operators with a certain harmonic symbol on the disk in C. We further extended the same problem to the weighted Bergman space and obtained the analogous result as in the unweighted Bergman space. This thesis will also survey the positivity of Toeplitz operators with bounded and unbounded symbols on the Bergman, Fock and certain Model spaces in terms of the Berezin transform of the symbol. Inspired by the paper of Zhao and Zheng, we have studied positivity of the Toeplitz operators with a bounded symbol on the Model spaces in terms of the Berezin transform. Analogous results have been obtained in the Fock space case as well.