Needle insertion is a critical aspect of many medical treatments, diagnostic methods, and scientific studies, and is considered to be one of the simplest and most minimally invasive medical procedures. Robot-assisted needle steering has the potential to improve the effectiveness of existing medical procedures and enable new ones by allowing increased accuracy through more dexterous control of the needle tip path and acquisition of targets not accessible by straight-line trajectories. In this article, we describe a robot-assisted needle steering system that uses three integrated controllers: a motion planner concerned with guiding the needle around obstacles to a target in a desired plane, a planar controller that maintains the needle in the desired plane, and a torsion compensator that controls the needle tip orientation about the axis of the needle shaft. Experimental results from steering an asymmetric-tip needle in artificial tissue demonstrate the effectiveness of the system and its sensitivity to various environmental and control parameters. In addition, we show an example of needle steering in ex vivo biological tissue to accomplish a clinically relevant task, and highlight challenges of practical needle steering implementation.