  The irrationality exponent of an irrational number $\xi$, which measures the
approximation rate of $\xi$ by rationals, is in general extremely difficult to
compute explicitly, unless we know the continued fraction expansion of $\xi$.
Results obtained so far are rather fragmentary, and often treated case by case.
In this work, we shall unify all the known results on the subject by showing
that the irrationality exponents of large classes of automatic numbers and
Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes
contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the
Fermat numbers, the regular paperfolding numbers, which have been previously
considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new
classes such as the Stern numbers and so on. Among other ingredients, our
proofs use results on Hankel determinants obtained recently by Han.
