Modern-day cloaking technology, who would have guessed? Granted, current techniques only render objects invisible to a single wavelength (or narrow range of wavelengths) of light, so true invisibility is still a few steps away, but the potential is there - really, truly, almost tangibly within the grasp of modern science. One fun question now is whether an invisible person would be able to see. The question is nothing new - anyone with an understanding of how light and vision relate has asked it every time science fiction presents an invisible character - man or otherwise.
The answer has to depend on certain assumptions, and the problem can be tackled from different directions. First things first, however - a brief review to make sure the reader knows the basics of sight: light strikes an object, some wavelengths of light may or may not be absorbed, the rest are reflected off in various directions. Some of the reflected light reaches a person's eye, and strikes the retina, where it is absorbed. The energy of the absorbed light (which is inversely proportional to wavelength) is registered and converted into a signal to the brain that conveys the information as colors. Based on this, the requirement for sight is twofold - light must reach the retina of the eye, and it must be absorbed.
Invisibility could be achieved in different ways. The classic idea is to make a person physically transparent so that light would pass through them the same as it does through glass or air. Despite being transparent, glass is still visible to the eye, so it is not truly invisible. The reason for this is that it has a different refractive index than the surrounding air. At the interface between air and glass, light is refracted (bent), creating a boundary our eyes can see. True invisibility by transparency requires a near exact match of refractive indices. Want to make glass disappear? Try dunking a piece of pyrex glass into a container filled with corn oil. The corn oil and the pyrex (which is borosilicate glass) have almost identical indices of refraction, and the glass seems to vanish. Something or someone with the same index of refraction as air would disappear in the same way.
The problem with transparency is that if light passes through the invisible person, the light also passes through the retina without being absorbed. If the light is not absorbed, the person cannot see. If the light reaching the retinas were absorbed, allowing the person to see, then the retinas could not be fully transparent, and could be seen by others, who would undoubtedly be creeped out by the pair of floating hemispheres wandering around at eye level. With some effort, it is possible that retinas (either biological or artificial) could be developed that would absorb, process, and then re-emit the light in the same direction it was heading. This isn't unlike camoflauge technology, where cameras or screens depict the image on the other side of an object. There would still be a detectable distortion if someone was watching, but it would be less obvious than a naked retina.
Some materials are only transparent to certain wavelengths of light. Normal glass, for example, lets visible light through, but blocks ultra-violet. If a retina could be designed that was completely transparent at visible wavelengths, but opaque to infrared or ultraviolet light, then the invisible person could see objects illuminated at those wavelengths. The person wouldn't see color as the rest of the world did, but objects would still be distinguishable. The invisible person could navigate while remaining hidden.
Current technology creates "invisibility" more along the lines described by Star Trekkian cloaking technology. Light is made to curve around the object, returning to its original path once safely on the other side. If this could be done perfectly, at all visible wavelengths, all light would be diverted and never come into contact with the now "invisible" object. The problems for vision match, in a large part, those experienced by our transparent person. If no light reaches his (or her) retinas, they can't see. If some light is allowed to be absorbed by the retinas, they become observable. If some wavelengths outside of the visible spectrum are allowed to pass through, rather than being deflected around, then those can be exploited for "sight". The nice thing about an arrangement like this is that whatever objects are within the protected space are invisible, no matter what they are made from, simply because light is not reaching them. That means anything can be built inside, including a radar, for example, which would make for excellent "vision", so long as radio waves could pass the barrier. For that matter, equipment that could monitor sound reflections (sonar - like bats) could also provide imagery. In short, anything other than visible light could be used in place of traditional vision.
The short answer appears to be yes, a truly invisible person can see, but only if they are willing to replace traditional colors with other options. Such an invisible man could certainly discern the figure of a woman, and even the cut of her dress, but would be unable to tell you later on that she was a strawberry blonde with vivid green eyes, or that the dress was a blue as deep as the sea. Vision is indeed a possibility for the invisible person, but it would present an entirely different view of the world.