  This paper introduces an extension of the linear least-squares (or
Lomb-Scargle) periodogram for the case when the model of the signal to be
detected is non-sinusoidal and depends on unknown parameters in a non-linear
manner. The attention is paid to the problem of estimating the statistical
significance of candidate periodicities found using such non-linear
periodograms. This problem is related to the task of quantifying the
distributions of maximum values of these periodograms. Based on recent results
in the mathematical theory of extreme values of random field (the generalized
Rice method), we give a general approach to find handy analytic approximation
for these distributions. This approximation has the general form $e^{-z}
P(\sqrt z)$, where $P$ is an algebraic polynomial and $z$ being the periodogram
maximum.
  The general tools developed in this paper can be used in a wide variety of
astronomical applications, for instance in the studies of variable stars and
extrasolar planets. For this goal, we develop and consider in details the
so-called von Mises periodogram: a specialized non-linear periodogram where the
signal is modelled by the von Mises periodic function $\exp(\nu \cos \omega
t)$. This simple function with an additional non-linear parameter $\nu$ can
model lightcurves of many astronomical objects that show periodic photometric
variability of different nature. We prove that our approach can be perfectly
applied to this non-linear periodogram.
  We provide a package of auxiliary C++ programs, attached as the online-only
material. They should faciliate the use of the von Mises periodogram in
practice.
