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Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form.

A simple modification of the surface tension in the axisymmetric case leads to analogues of the Delaunay surfaces. Here we have derived an explicit parameterization of the most simple case of this new class of surfaces which can be considered as a generalization of the catenoids. The geometry of these surfaces depends on two real parameters and has been studied in some detail