  We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials
$\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier
series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion
over compact subsets of the complex plane. This is used to determine explicit
estimates on the constants in the approximation, and also to analyze
oscillatory phenomena which arise in certain cases.
  These results are transferred to the Apostol-Euler polynomials
$\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the
Apostol-Bernoulli polynomials.
