  Given an orthogonal polynomial sequence on the real line, another sequence of
polynomials can be found by composing these polynomials with a general M\"obius
transformation. In this work, we study the properties of such
M\"obius-transformed polynomials. We show that they satisfy an orthogonality
relation in given curve of the complex plane with respect to a varying weight
function and that they also enjoy several properties common to the orthogonal
polynomial sequences on the real line --- e.g. a three-term recurrence
relation, Christoffel-Darboux type identities, their zeros are simple, lie on
the support of orthogonality and have the interlacing property, etc. Moreover,
we also show that the M\"obius-transformed polynomials obtained from classical
orthogonal polynomials also satisfy a second-order differential equation, a
Rodrigues' type formula and generating functions. As an application, we show
that Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all
related to each other by a suitable M\"obius transformation. New orthogonality
relations for Bessel and Romanovski polynomials are also presented.
