For / a nonconstant meromorphic function on A = {|z|< 1} and r £E (inf l/l, sup j/]), let £(/, r)=(z6¿: \f(z)\= r). In this paper, we study the components of A\£(/, r) along with the level sets £(/, r). Our results include the following: If/ is an outer function and ñ a component of A\£(/, r), then 52 is a simply-connected Jordan region for which (hü) n (|z|= 1} has positive measure. If/and g are inner functions with £(/,/■) = £(g, s), then g = tj/™, where |i)|= 1 and a > 0. When g is an

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... ry meromorphic function, the equality of two pairs of level sets implies that g = cf, where c # 0 and a £ (-oo, oo). In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.