  We introduce a new approximation technique into the context of complex
dynamics that allows us to construct examples of transcendental entire
functions with unbounded wandering domains. We provide examples of entire
functions with an orbit of unbounded fast escaping wandering domains, answering
a long-standing question of Rippon and Stallard. Moreover, these examples cover
all possible types of simply connected wandering domains in terms of
convergence to the boundary. In relation to a conjecture of Baker, it was
unknown whether functions of order less than one could have unbounded wandering
domains. For any given order greater than $1/2$ and smaller than $1$, we
provide an entire function of such order with an unbounded wandering domain.
