Hyperbolic dynamical systems carry many invariant, ergodic, probability measures known as equilibrium states, and these measures have associated dynamical quantities, such as their entropies, dimensions, and Lyapunov exponents. There are a number of now classical rigidity results, which state that corresponding quantities for a priori different measures agree if and only if the measures agree, and the maps in consideration are exceptional in some sense. More recently there have been results in a complementary flexibility direction, which show that the only restrictions relating these values for non-exceptional systems are natural ones.