This paper is concerned with continuous-time pursuit and evasion games.
Typically, we have a lion and a man in a metric space: they have the same
speed, and the lion wishes to catch the man while the man tries to evade
capture. We are interested in questions of the following form: is it the case
that exactly one of the man and the lion has a winning strategy?

As we shall see, in a compact metric space at least one of the players has a
winning strategy. We show that, perhaps surprisingly, there are examples in
which both players have winning strategies. We also construct a metric space in
which, for the game with two lions versus one man, neither player has a winning
strategy. We prove various other (positive and negative) related results, and
pose some open problems.