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In this paper, we study an active IRS-aided simultaneous wireless information and power transfer (SWIPT) system. Specifically, an active IRS is deployed to assist a multi-antenna access point (AP) to convey information and energy simultaneously to multiple single-antenna information users (IUs) and energy users (EUs). Two joint transmit and reflect beamforming optimization problems are investigated with different practical objectives. The first problem maximizes the weighted sum-power harvested by the EUs subject to individual signal-to-interference-plus-noise ratio (SINR) constraints at the IUs, while the second problem maximizes the weighted sum-rate of the IUs subject to individual energy harvesting (EH) constraints at the EUs. The optimization problems are non-convex and difficult to solve optimally. To tackle these two problems, we first rigorously prove that dedicated energy beams are not required for their corresponding semidefinite relaxation (SDR) reformulations and the SDR is tight for the first problem, thus greatly simplifying the AP precoding design. Then, by capitalizing on the techniques of alternating optimization (AO), SDR, and successive convex approximation (SCA), computationally efficient algorithms are developed to obtain suboptimal solutions of the resulting optimization problems. Simulation results demonstrate that, given the same total system power budget, significant performance gains in terms of operating range of wireless power transfer (WPT), total harvested energy, as well as achievable rate can be obtained by our proposed designs over benchmark schemes (especially the one adopting a passive IRS). Moreover, it is advisable to deploy an active IRS in the proximity of the users for the effective operation of WPT/SWIPT.