  The nonlinear Hall effect, which is the second-order harmonic charge Hall
effect from the Berry curvature dipole in momentum space, has received much
attention recently. As the responses to higher harmonics of the driving ac
electric field are prominent and measurable, we develop a general nonlinear
theory by taking the charge and spin currents as well as the longitudinal and
transverse effects into account. We introduce the expansion order of the
electric field and Berry curvature multipole moment, where the Berry curvature
dipole is a particular one manifesting itself at the second harmonic order and
the second expansion order of the electric field. There are four cases with
conserving or breaking the time-reversal symmetry (TRS) and inversion symmetry
(IS). We find a specific ``selection rule" that only longitudinal odd harmonic
order charge currents exist for conserving both the TRS and IS, and with
breaking both symmetries, all harmonic order charge and spin currents are
nonzero. With conserving TRS and breaking IS, the charge Hall current exists at
even harmonic order, and the longitudinal charge current occurs at odd harmonic
order. Only the longitudinal spin current survives at even harmonic order. With
breaking TRS and conserving IS, only odd harmonic order charge and spin
currents can appear. Moreover, we observe that every harmonic order current
contains a series of infinite-order expansion of the electric field. We further
show that the Berry curvature dipole and quadrupole can be determined by
measuring the second and fourth harmonic order currents in experiments. This
may open a way to explore the higher responses of an ac driving system.
