Except the Haar wavelet which is not continuous, it is well-known in the literature that compactly supported dyadic orthonormal real-valued wavelets cannot have symmetry. In order to achieve symmetry property of wavelets as well as other desirable properties such as translation invariance and directionality, complex-valued wavelets have been studied in the literature and have found to be useful in several applications such as signal denoising and image processing. Compactly supported orthonormal complex wavelets with symmetry, smoothness and vanishing moments are of interest in both theory and application. However, it remained open whether there exist symmetric dyadic orthonormal complex wavelets with compact support and arbitrarily high smoothness, as Daubechies did in her celebrated paper \cite{Daub:orth} for the Daubechies dyadic orthonormal real-valued wavelets with compact support and arbitrarily high smoothness (but such real-valued wavelets lack symmetry). In this paper, we shall affirmatively answer this question by explicitly constructing a family of symmetric dyadic orthonormal complex wavelets with compact support, arbitrarily high smoothness and vanishing moments. Such symmetric orthonormal complex wavelets have a close and interesting connection to the Daubechies orthonormal real-valued wavelets. In addition, we also propose another family of symmetric dyadic orthonormal complex wavelets with compact support and arbitrarily high vanishing moments such that their associated orthonormal dyadic refinable functions possess some desirable vanishing moment property, namely, they have the desirable coiflet property. The method presented in this paper to construct symmetric dyadic orthonormal complex wavelets is explicit and constructive. Some examples of symmetric orthonormal complex wavelets will be provided to illustrate the results in this paper. We also explore the close relation between symmetric orthonormal complex wavelets and nonsymmetric orthonormal real-valued wavelets, as well as their connections to the problem of expressing a polynomial as a sum of squares of two polynomials.