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We present integrated antiresonant reflecting optical (ARROW) structures with hollow cores as a new paradigm for integrated optics with gases and liquids. ARROW waveguides with micrometer-sized hollow cores allow for single-mode propagation in low-index nonsolid core materials where conventional index guiding is impossible. Fabrication methods and considerations are described, and measurements of the core-size dependence of the waveguide loss are presented. We analyze the dependence of waveguide loss in hollow-core ARROW waveguides on polarization, shape, and wavelength. We propose two-dimensional hybrid device geometries using solid and nonsolid core ARROWs and show that they can form the basis of integrated fluorescence and Raman sensors. We derive the design principles for simultaneous low-loss propagation in both waveguides and efficient cross-coupling between waveguides.