We consider the most basic visibility-based pursuit-evasion problem defined as follows: Given a polygonal region, a searcher with 360° vision, and an unpredictable intruder that is arbitrarily faster than the searcher, plan the motion of the searcher so as to see the intruder. In this paper, we present simple necessary and sufficient conditions for a polygon to be searchable, which settles a decade-old open problem raised in [13]. We also show that every searchable polygon is also searchable by a searcher with two flashlights (that is, two ray visions). This implies, combined with the previous work [7], that there is an O(n ²)-time algorithm for constructing a search path for an n-sided polygon.