This paper provides a practical guide to wavelet analysis, with examples from time series of the El Niño–Southern Oscillation (ENSO). It compares wavelet analysis to the windowed Fourier transform, discusses the choice of wavelet basis functions, edge effects, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed using theoretical wavelet spectra for white and red noise processes. These tests help establish significance levels and confidence intervals for wavelet power spectra. Smoothing in time or scale can increase confidence in wavelet spectra. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis, such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence, are described. The statistical significance tests are used to quantify changes in ENSO variance on interdecadal timescales. Using new datasets extending back to 1871, the Niño3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880–1920 and 1960–90, and lower power during 1920–60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2–8-yr wavelet power in both longitude and time. Wavelet analysis is a tool for analyzing localized variations of power within a time series. It decomposes a time series into time–frequency space, allowing determination of dominant modes of variability and how those modes vary in time. The wavelet transform has been used in various geophysical studies. The paper provides a step-by-step guide to wavelet analysis, including statistical significance testing. The consistent use of ENSO examples adds to the ENSO literature and allows greater confidence in previous wavelet-based ENSO results. The use of new datasets with longer time series permits a more robust classification of interdecadal changes in ENSO variance. The paper describes the datasets used for the examples, the method of wavelet analysis, and includes a discussion of different wavelet functions. It presents theoretical wavelet spectra for white-noise and red-noise processes, compares them to Monte Carlo results, and uses them to establish significance levels and confidence intervals for the wavelet power spectrum. It describes time or scale averaging to increase significance levels and confidence intervals. It also describes other wavelet applications such as filtering, the power Hovmöller, cross-wavelet spectra, and wavelet coherence. The summary contains a step-by-step guide to wavelet analysis.This paper provides a practical guide to wavelet analysis, with examples from time series of the El Niño–Southern Oscillation (ENSO). It compares wavelet analysis to the windowed Fourier transform, discusses the choice of wavelet basis functions, edge effects, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed using theoretical wavelet spectra for white and red noise processes. These tests help establish significance levels and confidence intervals for wavelet power spectra. Smoothing in time or scale can increase confidence in wavelet spectra. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis, such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence, are described. The statistical significance tests are used to quantify changes in ENSO variance on interdecadal timescales. Using new datasets extending back to 1871, the Niño3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880–1920 and 1960–90, and lower power during 1920–60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2–8-yr wavelet power in both longitude and time. Wavelet analysis is a tool for analyzing localized variations of power within a time series. It decomposes a time series into time–frequency space, allowing determination of dominant modes of variability and how those modes vary in time. The wavelet transform has been used in various geophysical studies. The paper provides a step-by-step guide to wavelet analysis, including statistical significance testing. The consistent use of ENSO examples adds to the ENSO literature and allows greater confidence in previous wavelet-based ENSO results. The use of new datasets with longer time series permits a more robust classification of interdecadal changes in ENSO variance. The paper describes the datasets used for the examples, the method of wavelet analysis, and includes a discussion of different wavelet functions. It presents theoretical wavelet spectra for white-noise and red-noise processes, compares them to Monte Carlo results, and uses them to establish significance levels and confidence intervals for the wavelet power spectrum. It describes time or scale averaging to increase significance levels and confidence intervals. It also describes other wavelet applications such as filtering, the power Hovmöller, cross-wavelet spectra, and wavelet coherence. The summary contains a step-by-step guide to wavelet analysis.