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 problem of estimating the statistical significance of candidate periodicities found using such non-linear periodograms is examined. This problem is related to the task of quantifying the distributions of the maximum values of these periodograms. Based on recent results in the mathematical theory of extreme values of a random field (the generalized Rice method), a general approach is provided to find a useful analytic approximation for these distributions. This approximation has the general form e^{-z} P(√{z}), where P is an algebraic polynomial and z is the periodogram maximum. The general tools developed in this paper can be used in a wide variety of astronomical applications, for instance in the study of variable stars and extra-solar planets. With this in mind, we develop and cons