Neural networks (NNs) with intensive multiplications (e.g., convolutions and transformers) are powerful yet power hungry, impeding their more extensive deployment into resource-constrained edge devices. As such, multiplication-free networks, which follow a common practice in energy-efficient hardware implementation to parameterize NNs with more efficient operators (e.g., bitwise shifts and additions), have gained growing attention. However, multiplication-free networks in general under-perform their vanilla counterparts in terms of the achieved accuracy. To this end, this work advocates hybrid NNs that consist of both powerful yet costly multiplications and efficient yet less powerful operators for marrying the best of both worlds, and proposes ShiftAddNAS, which can automatically search for more accurate and more efficient NNs. Our ShiftAddNAS highlights two enablers. Specifically, it integrates (1) the first hybrid search space that incorporates both multiplication-based and multiplication-free operators for facilitating the development of both accurate and efficient hybrid NNs; and (2) a novel weight sharing strategy that enables effective weight sharing among different operators that follow heterogeneous distributions (e.g., Gaussian for convolutions vs. Laplacian for add operators) and simultaneously leads to a largely reduced supernet size and much better searched networks. Extensive experiments and ablation studies on various models, datasets, and tasks consistently validate the effectiveness of ShiftAddNAS, e.g., achieving up to a +7.7% higher accuracy or a +4.9 better BLEU score as compared to state-of-the-art expert-designed and neural architecture searched NNs, while leading to up to 93% or 69% energy and latency savings, respectively. Codes and pretrained models are available at https://github.com/RICE-EIC/ShiftAddNAS.