The proof of the index formula of the Toeplitz operator with a continuous symbol on the Hardy space for the unit circle in the complex plane depends on the Hopf theorem. However, the analogue result of the Hopf theorem does not hold on a general connected domain. Hence, the extension of the index formula of the Toeplitz operator on a general domain needs a method which is different from that for the case of the unit circle. In the present paper, the index formula of the Toeplitz operator with a continuous symbol on the finite complex connected domain in the complex plane is obtained, and the cohomology groups of Toeplitz algebras on general domains are discussed. In addition, the Toeplitz operators with symbols in QC are also discussed.