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Ask most anybody about dogs or cats, and they will repeat the claim that these animals are colorblind. To most, it means greyscale. Is it true that cats (ref 2) and dogs (ref 2, ref 3) only see in grayscale?
I've noticed that my cats somehow don't seem to see very well, although at night they play with each other without any light at all.
TL;DR: The claim is actually false. Cats and Dogs (as well as many other animals) just see the world differently from humans. We would call them colourblind, but not greyscale colorblind.
I run into this claim A LOT, so I feel that it is indeed a great skeptical claim. At the very least a teaching moment to introduce what skepticism means since this is such a common belief which people can change because their worldview isn't tied too much to animals seeing in greyscale versus dicromatic scales. Okay, on to the answer.
Overall, a great explanation of how human color vision came to be can be found on this University of California Santa Cruz site. In particular, it talks about the Opsin evolution of trichromatic sight. Of particular note is that evolutionarily, trichormatic and dichromatic sights has gone back and forth a few times. In particular as shown by this diagram (wikicommons license):
Okay, all this lead up material, to simply answer your question, dogs and cats are not greyscale seeing animals. They are simply red-green color blind (so this may explain why they will ignore a yellow tennis ball on green grass unless they can locate it by scent). Dr. Mark Plonsky at the University of Wisconsin at Steven-Point gives a great rundown.
Dogs are red-green color blind. They see a brighter and less detailed world when compared to humans. Peripheral vision is better than humans (dogs see more of the world), but distance is not judged quite as well. Dogs excel at night vision and the detection of moving objects. Figure 1 is a rough guesstimate of what a dog and human might see when viewing a color band (the electromagnetic spectrum).
Cornell University also answers this question. Note that horses see things differently from cats and dogs.
Many adults think that dogs and cats cannot see any colors, only shades of gray. But this is not true. Cats and dogs can see some colors but not all of them. So in a sense they are like adults who are colorblind. Cats and dogs have cells in their eyes that respond well in bright and dim light. At night, cats and dogs use cells called rods that are sensitive to dim light. They are found in a lining at the back of the eye called the retina. When light falls on the rods, they send a message to the brain to explain the image that they see. For the bright light of day, you need cells called cones. But having cones in your eyes also means that you can also see color. Humans have three kinds of cones that allow them to see blue, red and green. So humans (and monkeys!) can see in full color. Dogs and cats only have two kinds of cones sensitive to blue and green light. So they do see some colors. By the way, if you have a horse or pony at home, they have red and blue cones. Horses see some colors, but they can't tell green from gray.
I hope that is a good rundown.
ucsc.edu reference says that abilities are gained and lost, and that the 'first mammal' was certainly trichromatic and even tetrachromatic for a time, but I don't understand it well enough to get any further details from it.
Cats and dogs have what we would call red-green color blindness. But they don't see only in grayscale.
Dogs and cats are dichromats (having two types of color receptors). (Neitz et al. 1989, Loop et al. 1979) Their cones allow them to distinguish between blue and yellow wavelengths, but less well between red and green wavelengths.
In cats, the two cone types have peak responses to wavelengths of 445-455 nm (deep blue) and 555 nm (greenish yellow). (Jacobs 1993, at p. 429)
In dogs, these receptors have peak responses to wavelengths of 429 nm (blue) and 555 nm (greenish yellow). (Neitz et al. 1989, Jacobs 1993 at p. 427)
Jacobs, G. H. (1993). The distribution and nature of colour vision among the mammals. Biological Reviews, 68(3), 413-471.
Loop, M. S., Bruce, L. L., & Petuchowski, S. (1979). Cat color vision: The effect of stimulus size, shape and viewing distance. Vision research, 19(5), 507-513.
Neitz, J., Geist, T., & Jacobs, G. H. (1989). Color vision in the dog. Vis Neurosci, 3(2), 119-125.