In this talk, we discuss the spectrum of the twisted Laplacian operator on a compact hyperbolic surface. The twisted Laplacian associated with a harmonic form is obtained by conjugating the usual Laplacian-Beltrami operator by an integral of the harmonic form. It is also the Bochner Laplacian associated with the corresponding one-dimensional representation. When the harmonic form is real, the twisted Laplacian is non-self-adjoint but still has discrete spectrum in the complex plane. We will review its spectral theory and connection with the twisted Selberg zeta function. In particular, we show that although most of the spectrum is concentrated near the real axis, there are infinite many eigenvalues away from the real axis, at least when the harmonic form is large enough. This implies the failure of the asymptotic version of Riemann hypotheses for the twisted Selberg zeta function as well as the failure of quantum unique ergodicity. This is joint work with Gong Yulin.