How to export a layer or artboard as SVG
File > Export in Paint by Zaweba writes shapes, gradients, strokes, paths and text as outlines to SVG, with optional pixel layers included as PNG.

Exporting to SVG is done from File > Export in Paint by Zaweba, which writes your shape layers, document paths and text to an SVG file. Solid and gradient fills, strokes, dashes and stroke profiles come across, text is written as outlines so it looks right without the font, and pixel layers can be included as PNG if you want them. There's also Paths to SVG for sending just the paths.
SVG is the format websites, cutting machines and most design programs read for sharp, scalable artwork. A logo exported once as SVG can be shown at any size without going blurry.
What goes into the file
Everything that is vector in Paint goes across as vector: shapes from the Shape and Pen tools, their fills and strokes, dashed lines, and the stroke profiles that taper a line. Gradient fills stay gradients rather than becoming pictures. Text becomes outline paths, which means the person opening the file sees exactly your font even if they don't own it, though they can't then edit the words.
Pixel layers are optional. Include them and they are embedded as PNG images inside the SVG, which is useful for a design that mixes a photo with vector type. Leave them out for a pure vector file.
Some things don't travel. Layer Art effects, masks and blend modes are left out, because SVG can't hold them in the same way. If your logo relies on a drop shadow from Layer Art, it won't be in the SVG. Either rebuild the shadow as a shape, or export a PNG for that use and keep the SVG for the plain mark.
Getting a clean export
Before you export, tidy the layers. Hide anything you don't want, such as guides, notes or rough sketches. Merge boolean results with Merge Shape Components so the file contains clean single paths rather than stacks of overlapping pieces. Check the artboard size, since that sets the size of the SVG.
If you need the text to stay editable for someone else, send them the .ptx file or a note of the font as well, because the SVG will carry outlines only. For web use that's usually what you want anyway.
Bringing SVG back in
SVG works the other way too. When you open an SVG, Paint asks for a width and whether to bring it in as pixels or as editable shape layers. Choose shape layers to carry on editing a logo, or pixels when you just want to place a vector icon on a photo at a set size.
When SVG is the wrong choice
A photograph saved as SVG is just a PNG inside a wrapper, so it gains nothing. Use JPEG, PNG or WebP for photographs and painted artwork. SVG shines for logos, icons, diagrams, lettering and anything that was built from shapes. If you have a bitmap logo and want a real vector version, Vector > Vectorise Bitmap traces it into shape layers first, and those export cleanly.
Paint by Zaweba is a free, offline image editor with vector shapes and type built in, so you can design and export an SVG without a separate drawing program.