We review recent developments and current status of liquid-core optical waveguides in optofluidics with emphasis on suitability for creating fully planar optofluidic labs-on-a-chip. In this first of two contributions, we give an overview of the different waveguide types that are being considered for effectively combining micro and nanofluidics with integrated optics. The large number of approaches is separated into conventional index-guided waveguides and more recent implementations using wave interference. The underlying principle for waveguiding and the current status are described for each type. We then focus on reviewing recent work on microfabricated liquid-core antiresonant reflecting optical (ARROW) waveguides, including the development of intersecting 2D waveguide networks and optical fluorescence and Raman detection with planar beam geometry. Single molecule detection capability and addition of electrical control for electrokinetic manipulation and analysis of single bioparticles are demonstrated. The demonstrated performance of liquid-core ARROWs is representative of the potential of integrated waveguides for on-chip detection with ultrahigh sensitivity, and points the way towards the next generation of high-performance, low-cost and portable biomedical instruments.