We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that the Thue-Morse word has the minimum possible subword complexity over all overlap free binary words and all (7/3)-power-free binary words, but not over all (7/3)(+)-power-free binary words;
the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all (7/3)-power-free binary words;
if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word;
the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words.