The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface X_C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface X_C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface X_C is called symmetric and the curve C is real.