Statistical properties of quantum transport are considered for a chaotic cavity with an arbitrary number of open channels. In the framework of the random matrix approach, we establish the relevance of the Selberg integral theory to the problematic and apply it to calculate exact explicit expressions of low-order cumulants of the conductance and shot-noise. By further exploiting the marriage of the Selberg integral with the theory of symmetric functions (Schur functions), we develop a powerful method for computing the moments of the conductance and shot-noise (including their joint moments) of arbitrary order and at any number of open channels. The approach is applicable equally well for systems with and without time-reversal symmetry. We also give a detailed discussion of the corresponding cumulants, the distribution functions, etc.