Three groups of guidewire embodiments are described with particularly suitable structures for navigating circuitous vessels, especially blood vessels of the brain. Some of the guidewires have a hyperbolic taper that provides desired flexibility. In some embodiments, an integrated guide structure provides for extension in the blood vessel of a corewire to provide for extended reach of the guidewire. In further embodiments, the guidewire has a flexible tip that can be guided directly by the flow in the vessel.
