  The Ap\'ery polynomials and in particular their asymptotic behavior play an
essential role in the understanding of the irrationality of \zeta(3). In this
paper, we present a method to study the asymptotic behavior of the sequence of
the Ap\'ery polynomials ((B_{n})_{n=1}^{\infty}) in the whole complex plane as
(n\rightarrow \infty). The proofs are based on a multivariate version of the
complex saddle point method. Moreover, the asymptotic zero distributions for
the polynomials ((B_{n})_{n=1}^{\infty}) and for some transformed Ap\'ery
polynomials are derived by means of the theory of logarithmic potentials with
external fields, establishing a characterization as the unique solution of a
weighted equilibrium problem. The method applied is a general one, so that the
treatment can serve as a model for the study of objects related to the Ap\'ery
polynomials.
