The simplest way to clean up an acrylic or an oil painting on stretched canvas is to use a white cotton cloth soaked in a gentle soapy water (not too wet though since it may loosen some of the varnish). Olive oil–based soap is the soap that works bestir me. Be gentle with paintings with thick paint, as you might break hardened paint. You might want to use q-tips and work gently in crevasses. If the painting still looks grimy, it’s better to see an art restorer that would use a stronger art cleaning product and may reapply pigment colors where need be. You will see a huge difference just by removing the accumulation of cigarette or fireplace smoke.

One really important thing to consider is not exposing paintings to extreme temperatures or shifts of temperature, for example freezing cold or extreme hot next to a fireplace. The canvas may start losing its elasticity and the middle might give in to gravity. But even at normal room temperature, canvases may become loosened on the stretcher bars. A simple method to retighten the canvas is to spray water (again, don’t overdo it) on the back of the canvas and leave the piece to dry for a couple of hours. Canvases are made of fabric, and with time the weave will become loose.

I have repaired canvases for years spraying water on the back. Not only mine but others as well. When art is stored at galleries it is often in cold storage rooms which may have that effect on them. It is a safe process that will not damage the painting itself.

As for works on paper, you need to determine first if it is a water-based paint or not. I don’t use water-based paint usually but it happens once or twice a year. You can test to see if it’s water-based by wiping a damp white cloth in an area of the piece that would be behind a mat. If the color of the paint can be seen on the cloth, it is a water-based paint. If that’s the case then you would damage the piece while trying to clean using water. I would recommend to simply dust using a dry soft and fine brush. With non-water-based paint, a soft cleaning with a damp, barely humid white cotton cloth can do wonders. Avoid wetting the exposed paper without paint, as you could create a water stain.

Most of the time, a work on paper looks very grimy because of the accumulation of dirt on the glass itself, which could easily be cleaned up and the piece put back together. If that is the case then it is better to not clean the piece at all but clean the glass and maybe even refresh the frame by giving it a smooth polish with furniture cleaner.