A guidewire for medical use such as in vascular and nonvascular systems. The guidewire made from a titanium molybdenum alloy wire with a composition of approximately 78% titanium 11.5% molybdenum 6% zirconium and 4.5% tin by weight such that it is softer than stainless steel guidewires and stiffer than NiTi alloy guidewires. The distal end of the guidewire is of a smaller diameter and softer than the proximal end and fitted with a coil for springiness such that the distal end will bend when encountering curves in the body passageways. The distal tip may be heat treated for a gradient of softness with the distal tip being the softest. The distal end may also be tapered to provide an additional gradient of softness. A distal tip on the distal end of the guidewire protects the wall of the passageway from being punctured as the guidewire travels through the passageway. The resulting guidewire has properties between those of stainless steel guidewires and NiTi alloy guidewires for better torsion and stiffness characteristics.