What's the best way to implement `next` and `previous` on an enum type?

Try this:

public static enum A { 
    X, Y, Z;
    private static A[] vals = values();
    public A next()
    {
        return vals[(this.ordinal()+1) % vals.length];
    }

Implementation of previous() is left as an exercise, but recall that in Java, the modulo a % b can return a negative number.

EDIT: As suggested, make a private static copy of the values() array to avoid array copying each time next() or previous() is called.

Alternatively, one can go somehow along the lines of the following idea:

public enum SomeEnum {
  A, B, C;

  public Optional<SomeEnum> next() {
    switch (this) {
      case A: return Optional.of(B);
      case B: return Optional.of(C);
      // any other case can NOT be mapped!
      default: return Optional.empty();
  }
}

Notes:

1. In contrast to the other answer, this way does some implicit mapping; instead of relying on ordinal(). Of course that means more code; but it also forces the author to consider what it means to add new constants or remove existing ones. When relying on ordinal, your implicit assumption is that the order is based on the order used for the enum constant declaration. So when somebody comes back 6 months later and has to add a new constant, he has to understand that the new constant Y needs X, Y, Z ... instead of just appending X, Z, Y!
2. There might be situations where it doesn't make any sense for the "last" enum constant to have the "first" as successor. Think of T-Shirt sizes for examples. XXL.next() is for sure not XS. For such situations, using Optional is the more appropriate answer.