In this paper, we study the waveform and passive beamforming design for intelligent reflecting surface (IRS)-aided wireless power transfer (WPT). Generalized multi-user and low complexity single-user algorithms are derived based on alternating optimization (AO) framework to maximize the weighted sum output DC current, subject to transmit power constraints and passive beamforming phases unit modulus constraints. The input signal waveform and IRS passive beamforming phase shifts are jointly designed as a function of users' individual frequency-selective channel state information (CSI). The energy harvester nonlinearity is explored and two IRS deployment schemes, namely frequency selective IRS (FS-IRS) and frequency flat IRS (FF-IRS), are modeled and analyzed. This paper highlights the fact that IRS can provide an extra passive beamforming gain on output DC power over conventional WPT designs and significantly influence the waveform design by leveraging the benefit of passive beamforming, frequency diversity and energy harvester nonlinearity. Even though FF-IRS exhibits lower output DC current than FS-IRS, it still achieves substantially increased DC power over conventional WPT designs. Performance evaluations confirm the significant benefits of a joint waveform and passive beamforming design accounting for the energy harvester nonlinearity to boost the performance of single-user and multi-user WPT system.