Remarkable properties of optical fibers with a high-index core region and a surrounding silicarair photonic crystal cladding have recently been reported. Here we discuss the physics, the special guiding properties, and the theoretical tools developed for the modeling of these photonic crystal fibers. With an emphasis on the applicational aspects of the fibers, we study their single-mode operation, bending losses, and dispersion properties. While exhibiting certain unique properties, the high-index core photonic crystal fibers share many common features with conventional optical fibers, attributed to an operation based on the well-known mechanism of total internal reflection. Fundamentally different from all high-index core fibers, in this work we demonstrate a novel type of optical waveguide, operating truly by the photonic bandgap effect. The novel fiber has an improved photonic crystal cladding and a central low-index structural defect along which the light is guided. The novel fiber has several unique features due to its different waveguidance mechanism, including remarkable dispersion