Place each of the twelve pentominoes into the grid once, with rotations and reflections allowed. No two pentominoes can overlap or be orthogonally adjacent, and all cells not occupied by the pentominoes must be traversed by a single, closed loop (connecting cells horizontally or vertically). The black circles indicate some cells that must be covered by pentominoes, white circles indicate spaces that may not be contained in pentominoes and must be part of the loop.
Unless I made a mistake somewhere, this solution is unique:
To get there, I used a couple of rules of thumb:
1. Never make a separated area of empty squares
2. Never make a "dead end" for the empty square path
2a. The "U" piece comes with a dead end, so that piece must hug a knoll on the edge
3. Each piece must go somewhere.
4. If an empty square has only two (possibly) empty neighbours, the path must connect them.
Apart from those, there were a couple of slightly mind-boggly deductions required, but all in all, everything seemed extremely well designed, and no guesswork was needed at any point.
Progress, part 1:
Starting with the easiest bits, I make a (non-critical) mistake right off the bat. The red pieces all directly follow from the rules of thumb above, as I keep plotting the only possible path from the lower left corner. The mistake is that the blue bit on the right isn't necessarily connected all the way up. (Fixed that in some later image, when I spotted my mistake.)
Progress, part 2:
There must be a path of empty surrounding the pentomino on the right, and the neighbouring pentomino forces that path so much that the border piece must be the N-pentomino. There is no room for another pentomino in the lower right corner, so it must be empty. There's only one path that can achieve that.
Progress, part 3:
Counting from the bottom right corner, the empty square at (4,4) forces the square below it to also be empty. If it weren't empty, there would be a 3-way junction of the empty path on the right.
Also, the black dots right above the (4,4) must be connected: if they were not, there would be a dead end (or a disconnected empty space) between them.
Progress, part 4:
This bit is very tricky. On the bottom side, the two rightmost black circles must be connected, because if they weren't, the piece in the middle lower side would have to be the P. No matter which way you try to put the P, there will always be a dead end somewhere. Since the empty path blocks the other options, the piece on the lower side must be the Y. This, interestingly, forces the empty path shape quite a lot, even to the point where the P piece's place becomes forced.
Progress, part 5:
The pentomino on the left lower side can only be T, P or N. Since the P and N are already elsewhere, it must be the T. This squeezes the empty path towards the left, so the pentomino on the left side has to be the V. Also, at this point, there is only one spot where the U can go, so that the dead end in its middle is outside the borders.
It's definitely a bit difficult to spot, but at this point, there is exactly one possible spot for the X piece.
After that, there weren't any major difficulties in placing the rest of the pieces and finding the only possible path for the empty squares. Here's one more picture from along the way:
(Apologies for the inconsistent scaling of the images, Imgur's auto scaling seems to have malfunctioned on some of the pictures.)