Introduction. We say an operator T on a Hilbert space H is hyponormal if Tx || > || T*x || for xeH, or equivalently T*T-TT* > 0. In this paperwe will first examine some general properties of hyponormal operators. Then we restrict our interest to hyponormal operators with "thin" spectra. The importance of the topological nature of the spectrum is evident in our main result (Theorem 4) which states that a hyponormal operator whose spectrum lies on a smooth Jordan arc is normal. We continue with a general discussion of a certain growth condition on the resolvent which obtains for hyponormal operators. We conclude with a counterexample to a relation between hyponormal and subnormal operators. The reader is advised that additional facts about hyponormal operators may be found in [l1]. We shall denote the spectrum and the resolvent set of an operator by o(T) and p(T), respectively. The spectral radius R,,(T) sup {j z : z E a(T)}. The numerical range = closure {z: z Tx.x) 11 x = 1} is designated by W(T). Throughout the paper the underlyingvector space is always a separable Hilbert space H.