Singular spectrum analysis (SSA), a linear (univariate and multivariate) time series technique, performs principal component analysis (PCA) on an augmented data set containing the original data and time-lagged copies of the data. Neural network theory has meanwhile allowed PCA to be generalized to nonlinear PCA (NLPCA). In this paper, NLPCA is further extended to perform nonlinear SSA (NLSSA): First, SSA is applied to reduce the dimension of the data set; the leading principal components (PCs) of the SSA then become inputs to an NLPCA network (with a circular node at the bottleneck). This network performs the NLSSA by nonlinearly combining all the input SSA PCs. The NLSSA is applied to the tropical Pacific sea surface temperature anomaly (SSTA) field and to the sea level pressure anomaly (SLPA) field for the 1950--2000 period. Unlike SSA modes, which display warm and cool periods of similar duration and intensity, NLSSA mode 1 shows the warm periods to be shorter and more intense than the cool periods, as observed for the El Ni¿o-Southern Oscillation. Also, in SSTA NLSSA mode 1 the peak warm event is centered in the eastern equatorial Pacific, while the peak cool event is located around the central equatorial Pacific, an asymmetry not found in the individual SSA modes. A quasi-triennial wave of about a 39 month period is found in NLSSA mode 2 of the SSTA and of the SLPA.