A full-frame sensor can be an asset for shooting in low light, but it's not a requirement.
The general rule is that the bigger the sensor, the better equipped it is for low-light scenarios. This is because it has a bigger surface area so it can capture more light but it's not quite as simple as that. While there are a lot of full-frame cameras in our guide, cameras with back-illuminated sensors tend to perform best in low-light conditions. You also need to take into account the pixel count as cameras with a smaller pixel count use larger individual pixels so they are better at gathering light information.
That being said, you can get excellent APS-C cameras in low-light. You just need to weigh up what is most important to you in a camera system. Cropped sensor cameras are often smaller, lighter, more portable, and have smaller lenses so if you intend to travel a lot or have it on you at all times, it might be an idea to think about that too. There are also Micro Four Thirds cameras that perform well in low light, and even compacts with 1/2.3-inch sensors can do well in the right circumstances. We've included all of the above in our guide.