  We study Muttalib--Borodin ensembles --- particular eigenvalue PDFs on the
half-line --- with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We
show how the theory of the Selberg integral, involving also Jack and Schur
polynomials, naturally leads to a multi-parameter generalisation of these
particular Muttalib--Borodin ensembles, and also to the explicit form of
underlying biorthogonal polynomials of a single variable. A suitable
generalisation of the original definition of the Muttalib--Borodin ensemble
allows for negative eigenvalues. In the cases of generalised Gaussian,
symmetric Jacobi and Cauchy weights, we show that the problem of computing the
normalisations and the biorthogonal polynomials can be reduced down to
Muttalib--Borodin ensembles with classical weights on the positive half-line.
