This paper deals with continuous frames and continuous Riesz bases. We introduce continuous Riesz bases and give some equivalent conditions for a continuous frame to be a continuous Riesz basis. It is certainly possible for a continuous frame to have only one dual. Such a continuous frame is called a Riesz-type frame. We show that a continuous frame is Riesz-type if and only if it is a continuous Riesz basis. Finally we find a measure with respect to which, a continuous wavelet frame is a continuous Riesz basis.