Yes. The cathode is always the electrode where reduction occurs, and the anode is the electrode where oxidation occurs.
In a galvanic cell, a spontaneous chemical reaction gives rise to an electric current when connected. The reduction reaction at the cathode consumes electrons, making it positively charged, while the oxidation reaction at the anode releases electrons, making it negatively charged, so electrons flow through the external circuit from anode to cathode.
In an electrolytic cell, passage of electric current causes a non-spontaneous chemical reaction to occur. Electrons have to be pushed into the cathode to effect the reduction, and pulled out of the anode to effect the oxidation. Hence the cathode is the negative electrode, and the anode the positive electrode, while in a battery the cathode is the positive electrode. But in each case the cathode is where reduction happens - that's the important thing.