Accurate determination of pulse onset and peak is important in many photoplethysmogram (PPG) signal analysis applications. This paper presents a new automatic robust pulse onset and peak detection method which consists of the stationary wavelet transform; the multiscale sum and product; the adaptive amplitude thresholding; the smooth Shannon entropy envelope extraction; the Gaussian derivative filter-based peak finding; and the peak and onset determination and correction. The proposed method achieves an average sensitivity (Se) of 99.66\%, positive predictivity (Pp) of 99.90\%, and overall accuracy (OA) of 99.55\% on a total number of 1 16 255 beats taken from the Massachusetts Institute of Technology-Beth Israel Hospital polysomnographic sleep and complex system laboratory databases and finger pulse database. We further studied the robustness of three peak detection methods such as bandpass filter and Hilbert transform (BPF + HT) {[}12], pulse waveform delineator (PUD) {[}20], and the proposed method using the noisy PPG signals with a signal-to-noise ratio (SNR) ranging from 30 to 10 dB. The method achieved an average Se of 99.51\%, Pp of 99.92\%, and OA of 99.43\% for the PPG signals with SNR of 10 dB whereas the BPF + HT and PUD methods achieved an average Se = 99.66\%, Pp = 87.71\%, and OA = 87.45\%, and Se = 68.04\%, Pp = 70.38\%, and OA = 52.89\%, respectively. The proposed method significantly outperforms the other methods in terms of detection accuracy and robustness under noise-free and noisy PPG recordings. Unlike the existing methods, the proposed method does not use search-back algorithms to reject or include the noise peaks or missed peaks.