A guidewire kit comprising: a guidewire (1; 21; 61) for guidance of a catheter or of a catheter introducer sheath into a human patient, the guidewire (1; 21; 61), when in untensioned condition, having: - at its distal end, a rounded tip (4; 24) having a radius larger than a guidewire circumference radius or a floppy tip section having more flexibility than a proximally adjacent section of the guidewire; - a curved deflection section (2; 22) closely proximal of the tip (4; 24); and - a shaft section (3; 23) proximal of the curved deflection section (2; 22); wherein the curved deflection section (2; 22) extends over an angle of curvature such that the tip (4; 24) is spaced from a continuation of an axis of the shaft section (3; 23); and wherein the shaft section (3; 23) has a straight section (5; 25) proximally neighboring said deflection section (2; 22); and a straightener (34; 74) bounding a lumen for straightening at least the curved deflection section (2; 22) of the guidewire (1; 21; 61) extending into the lumen for insertion of the tip (4; 24) and the curved deflection section (2; 22) of the guidewire (1; 21; 61) into a needle inserted into a vessel (10; 31) of a patient; characterized in that, the shaft section (3; 23) has a curved steering section (6; 26) proximal of said straight section (5; 25); the straightener (34; 74) is of a length of at least 8 cm; the straightener lumen has a curved section (36; 76); and the curved steering section (6; 26) of the guidewire (1; 21; 61) is at least partially located in the curved section (36; 76) of the lumen of the straightener (34; 74) if the tip (4; 24) of the guidewire (1; 21; 61) projects from the straightener (34; 74) over at least one distance in a range between 5 and 15 cm. a guidewire (1; 21; 61) for guidance of a catheter or of a catheter introducer sheath into a human patient, the guidewire (1; 21; 61), when in untensioned condition, having: - at its distal end, a rounded tip (4; 24) having a radius larger than a guidewire circumference radius or a floppy tip section having more flexibility than a proximally adjacent section of the guidewire; - a curved deflection section (2; 22) closely proximal of the tip (4; 24); and - a shaft section (3; 23) proximal of the curved deflection section (2; 22); - at its distal end, a rounded tip (4; 24) having a radius larger than a guidewire circumference radius or a floppy tip section having more flexibility than a proximally adjacent section of the guidewire; - a curved deflection section (2; 22) closely proximal of the tip (4; 24); and - a shaft section (3; 23) proximal of the curved deflection section (2; 22); wherein the curved deflection section (2; 22) extends over an angle of curvature such that the tip (4; 24) is spaced from a continuation of an axis of the shaft section (3; 23); and wherein the shaft section (3; 23) has a straight section (5; 25) proximally neighboring said deflection section (2; 22); and a straightener (34; 74) bounding a lumen for straightening at least the curved deflection section (2; 22) of the guidewire (1; 21; 61) extending into the lumen for insertion of the tip (4; 24) and the curved deflection section (2; 22) of the guidewire (1; 21; 61) into a needle inserted into a vessel (10; 31) of a patient; characterized in that, the shaft section (3; 23) has a curved steering section (6; 26) proximal of said straight section (5; 25); the straightener (34; 74) is of a length of at least 8 cm; the straightener lumen has a curved section (36; 76); and the curved steering section (6; 26) of the guidewire (1; 21; 61) is at least partially located in the curved section (36; 76) of the lumen of the straightener (34; 74) if the tip (4; 24) of the guidewire (1; 21; 61) projects from the straightener (34; 74) over at least one distance in a range between 5 and 15 cm.