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Albeit the speed and energy advantages offered by silicon (Si) and silicon nitride (SiN) photonics, a single material platform, with varying intrinsic limitations, can hardly meet the ever-diversifying requirements posed for photonic integrated circuits (PICs). The heterogeneous integration of functional materials to established photonic platforms has thus gained tremendous momentum, underpinning various high-performance optoelectronic applications. However, stringent lattice-matching constraints severely hinder the heteroepitaxial material quality grown on dissimilar optical substrates. Here we obviate this hurdle by exploiting freestanding single-crystalline nanomembranes enabled by advanced epitaxy and layer lift-off techniques. We present a versatile framework leveraging photonic van der Waals (vdW) integration to infuse desired functionalities to Si and SiN photonics. By transferring single-crystalline thin-film barium titanate (BTO) to Si chips, we experimentally demonstrate ultraefficient electro-optical (EO) modulation, featuring giant Pockels coefficient r_42 over 1290 pm/V and a 3 dB EO bandwidth over 23 GHz, enabled by well-defined BTO crystallographic quality and orientations. Ultracompact non-reciprocal magneto-optical (MO) isolators are further realized, via vdW-integrating cobalt ferrite (CFO) nanomembranes to Si microrings, manifesting an ultrahigh Faraday rotation coefficient θ_F exceeding 33,800 °/cm. To embody the salient capabilities of coalescing functional materials with disparate crystal structures to arbitrary photonic templates, we laterally stitch gallium arsenide and gallium nitride single-crystals atop SiN photonics to realize expansive photodetection from ultraviolet to near infrared wavelengths. Multifunctional ring-resonators for simultaneous EO and MO modulations are also demonstrated by constructing vertical CFO/BTO heterostructures, ushering a vast playground to prototype record-setting heterogeneous integrated applications and study diverse physical coupling phenomena.