The paper concerns an important but underappreciated genre of algorithmic puzzles, explaining what these puzzles are, reviewing milestones in their long history, and giving two different ways to classify them. Also covered are major applications of algorithmic puzzles in cognitive science research, with an emphasis on insight problem solving, and the advantages of algorithmic puzzles over some other classes of problems used in insight research. The author proposes adding algorithmic puzzles as a separate category of insight problems, suggests 12 specific puzzles that could be useful for research in insight problem solving, and outlines several experiments dealing with other cognitive aspects of solving algorithmic puzzles.