  We determine the non-linear hydrodynamic response to geometrical fluctuations
in heavy ion collisions using ideal and viscous hydrodynamics. This response is
characterized with a set of non-linear response coefficients that determine,
for example, the $v_5$ that is produced by an $\epsilon_2$ and an $\epsilon_3$.
We analyze how viscosity damps both the linear and non-linear response
coefficients, and provide an analytical estimate that qualitatively explains
most of the trends observed in more complete simulations. Subsequently, we use
these nonlinear response coefficients to determine the linear and non-linear
contributions to $v_1$, $v_4$ and $v_5$. For viscous hydrodynamics the
nonlinear contribution is dominant for $v_4$, $v_5$ and higher harmonics. For
$v_1$, the nonlinear response constitutes an important $\sim 25%$ correction in
mid-central collisions. The nonlinear response is also analyzed as a function
of transverse momentum for $v_1$, $v_4$ and $v_5$. Finally, recent measurements
of correlations between event-planes of different harmonic orders are discussed
in the context of non-linear response.
