Consider a finite set of identical computational entities that can move freely in the Euclidean plane operating in Look-Compute-Move cycles. Let p(t) denote the location of entity p at time t; entity p can see entity q at time t if at that time no other entity lies in the line segment p(t)q(t). We consider the basic problem called Mutual Visibility: starting from arbitrary distinct locations, within finite time the entities must reach, without collisions, a configuration where they all see each other. This problem must be solved by each entity autonomously executing the same algorithm. We study this problem in the "luminous robots" model; in this generalization of the standard model of oblivious robots, each entity, called "robot", has an externally visible persistent light which can assume colors from a fixed set. The case where the number of colors is c=1 corresponds to the classical model without lights. In this paper we investigate under what conditions luminous robots can solve Mutual Visibility without collisions and at what cost (i.e., with how many colors). We establish a spectrum of results, depending on the power of the adversary, on the number c of colors, and on the a-priori knowledge the robots have about the system. Among such results, we prove that Mutual Visibility can always be solved without collisions in SSynch with c=2 colors and in ASynch with c=3 colors. If an adversary can interrupt and stop a robot moving to its computed destination, Mutual Visibility is still always solvable without collisions in SSynch with c=3 colors, and, if the robots agree on the direction of one axis, also in ASynch. All the results are obtained constructively by means of novel protocols. As a byproduct of our solutions, we provide the first obstructed-visibility solutions to two classical problems for oblivious robots: Collision-less Convergence to a point and Circle Formation.