1. Star Battle
Rows, columns and regions, as in Sudoku, with stars instead of digits. The closest relative with no numbers at all. You place stars so every row, column and region holds the same number, and they can never touch.
You can play it right here: today's Star Battle is free, or read the rules in a minute. It also goes by the names Two Not Touch and Queens.
2. Killer Sudoku
Sudoku with added sums. Standard Sudoku rules, plus cages of cells that must add up to a given total. It often starts with no digits on the board.
3. Kakuro
A crossword made of sums. Fill a crossword-shaped grid with digits so each run adds up to its clue, with no digit repeated in a run.
4. KenKen
Sudoku with arithmetic. Each row and column holds every digit once, and outlined cages must reach a target using a given operation.
5. Nonogram
Logic that draws a picture. Number clues tell you how many cells to fill in each row and column. When you finish, a picture appears.
6. Slitherlink
Logic with a line instead of digits. Draw one closed loop through a grid of dots, so each number shows how many of its sides the loop uses.
7. Hitori
Sudoku in reverse. Start with a full grid and shade cells until no number repeats in a row or column, without shaded cells touching.
8. Futoshiki
Sudoku with greater-than signs. Fill each row and column with every digit once, while obeying the inequality signs between cells.
Questions
What is the best game like Sudoku?
If you like the row, column and region logic, try Star Battle or Killer Sudoku. If you would rather avoid numbers, Star Battle and Nonograms are good places to start.
Are there Sudoku-like games without numbers?
Yes. Star Battle uses only stars, Nonograms use shaded cells to reveal a picture, and Slitherlink has you draw a loop. Nonograms and Slitherlink do have number clues, but you don't place digits.