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Why are textures always square powers of two? What if they aren’t?

I am using C# and XNA (also MonoGame for other platforms than Windows) to create a game in 2D. Ususally for all textures I use the NVidia plugin for Photoshop to create DDS files with precomputed mip-maps (either full color, or DXT-5).

How important is it, nowadays, to stick to textures with extensions that are a power of two? What are the advantages and disadvantages of doing so?

The background of my question for that I use a number of quite large background images in with 1980 x 1080 pixels in size and storing them in textures of 2048 x 2048 pixels size seems like a tremendous waste of memory.

Let me stress that I am aware that having only power-of-two textures has been important in the past for certain specific optimizations BUT I would like to know whether this TODAY (GPUs of the last ~2 years, current consoles etc.) is still required?