The grid is subdivided into multiple regions, all marked with thick lines.
Shade some cells so that unshaded cells form an orthogonally connected region. Shaded cells may not be orthogonally adjacent, and any orthogonal line of unshaded cells may straddle at most one thick line. A number indicates the precise number of shaded cells found in that region.
Some puzzles have special cells. A grey cell is a given shaded cell. A black cell is not part of the puzzle, and a cross is a given unshaded cell.
The position of a number within a region doesn't matter, and is typically the top-left cell of that region.
Puzzles in this genre
Consider the 3. There are two ways to shade 3 cells in this rectangle without them being adjacent, however one of these will trap an unshaded cell against the puzzle wall. As such, we can complete this rectangle in the only possible way.
Next, we can consider the top unshaded cell from that block and note that it needs to have a path of unshaded cells to join the others. As such, the top right region will be unshaded. Additionally we note that the right hand column of the top left region needs to be unshaded so as to allow the unshaded path to continue, and not have two adjacent shaded cells.
This leaves the top left region with a 2x2 square of potential cells to shade, and precisely 2 to shade. There are two ways of doing this, but again, one option leaves a trapped unshaded cell in the corner, so we can fill in the other.
Now considering the left hand edge, we have an unshaded cell which forces another unshaded cell beneath it. These two combined cross a thick black line and we are not allowed to have an unshaded straight path crossing two black lines, so we can deduce that the next cell down must be shaded.
Following the rest of the rules through quickly leads to no more remaining shaded cells.