Dominant strategy in chess
One of the following is true:


There is a dominant strategy for White.
There is a dominant strategy for Black.
There are strategies for both players that guarantee they don't lose, i.e. perfect play results in a draw (e.g. as in Tic-Tac-Toe).


No one knows which is true. Most experts guess that perfect play leads to a draw, and a few believe White can always win. It is unlikely Black has a dominant strategy, but no one has been able to rule it out either. Theoretically, a computer working for long enough could just try everything and tell us the answer, but with today's best technology and algorithms that would take prohibitively long.

Here are some important features of chess that together guarantee that one of the 3 cases above is true:


Chess is a game of complete information, i.e. both players always know everything that is going on.
Players take turns (as opposed to playing simultaneously, e.g. as in Rock Paper Scissors)
The game always ends eventually (at least under tournament rules as of 2014, after enough moves are made that aren't irreversibly advancing the game, it's a draw).
There is no randomness.
There are only 2 players.


Intuitively, these facts let you completely reason out the consequences of everything that could possibly happen (if I do A, my opponent could do B or C, in which case I could do D or E...). For a more formal explanation of why this implies there are either dominant or drawing strategies, see Zermelo's Theorem.

[Parts of this answer are synthesized from earlier answers/comments.]