A grooved stylet having a puncture point aligned with the stylet axis is mounted through a catheter to form an assembly. The aligned puncture point enables more accurate piercing of a vein, and reduces the possibility of double piercing the vein.The stylet near its proximal end is provided with a circular shoulder which abuts the proximal end of the catheter, and the catheter end is positioned on the stylet so that only the stylet puncture point projects beyond the catheter.The catheter is constructed of a hydrophilic polymer which expands away from the stylet and permits the stylet to be withdrawn from the puncture site, leaving the catheter in place in the vein.If desired, a plurality of bores may be provided in the catheter for separate attachment to different I.V. solutions; this enables various solutions to be fed at different rates to the patient through a single I.V. puncture site.An assembly of the catheter may be used either with a stylet or with a needle. In the latter case, the catheter may be used as an over-the-needle assembly, or a through-the-needle assembly.