Natural conditions are imposed on spectra of products and sums of operators. This results in characterizations of positive operators, Hermitian operators, compact operators, and unitary operators. Here are two main results: If S is an operator and the spectrum of ST consists of nonnegative real numbers for all invertible positive operators [noninvertible positive operators] T, then S is a positive operator. If S is an operator and the spectrum of ST is countable for all invertible operators [noninvertible operators] T, then S is a compact operator. The first half of the paper is primarily concerned with operators on finite-dimensional spaces, and the second half with operators on infinite-dimensional Hilbert spaces.