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Subdivision Surfaces
NURBS
Polygons

Subdivision Surfaces

RenderMan for Maya supports Maya subdivision surfaces (available in Maya Unlimited), including support for hierarchical editing, which allows for exact control over components of the subdivision mesh at finer levels of subdivision. Also, since RenderMan renders true curved surfaces, tessellation is never an issue. RenderMan handles subdivision surfaces well.

The subdivision surface is a relatively new geometric primitive which combines strengths of both NURBS and polygons.  These combined strengths make subdivision surfaces well suited for modeling complex objects, especially organic objects that require animation, like characters. 

A subdivision surface is described by a control mesh of points, like a NURBS surface. Additionally, a subdivision surface's control mesh is not confined to being rectangular, which is a major limitation of NURBS. In this respect, a subdivision surface's control mesh is analogous to a polygon model. But where polygon models require many facets to approximate being smooth, a subdivision surface is smooth; it is a true curved surface, meaning that Pixar's RenderMan actually renders the high-level subdivision surface (and only tessellates on a pixel basis). Subdivision surfaces will never have a faceted look (like polygon models can), no matter how the surface animates or how closely it is viewed.

A subdivision surface can be constructed to make highly efficient use of geometry, putting geometry only where it is needed — and not where it's not — a great benefit for animation.

Subdivision Surface Strengths:

1. A true curved surface (like NURBS) — with unique support for creases and points.
2. Animation — No patch cracks (like can happen with NURBS). Put geometry only where needed (unlike NURBS and polygons).
3. Displacements — High quality. (No cracks, a common artifact with displacements on polygons.)

With RenderMan, rendering a complex subdivision surface is much more efficient than rendering the equivalent polygon model (a model which would require many model facets to approximate a smooth surface).

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NURBS 

RenderMan for Maya provides complete support for Maya NURBS, including trimmed surfaces. NURBS (or nonuniform rational B-splines) are essentially rectangular 2D patches which are stretched and bent into 3D shapes. Because RenderMan renders true curved surfaces, tessellation is never an issue; NURBS are always smooth. 

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Maya uses U & V coordinates to parameterize NURBS surfaces, while RenderMan similarly uses S & T. RenderMan for Maya correctly interprets S & T values automatically. This would only be an issue when importing custom RenderMan shaders.

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Polygons

RenderMan for Maya provides full support for Maya polygons, including Maya's UV mapping tools. A single polygon is a face, with any number of edges and corners (vertices). A polygon mesh is composed of many individual polygons, a polygonal mesh. Polygons tend to be easier to model than NURBS since a polygonal mesh can be any arbitrary topology, whereas NURBS must be carefully organized patches. Polygons have several inherent disadvantages, however, when compared to NURBS or subdivision surfaces: 

1) Polygons are susceptible to faceting artifacts

2) Polygons require dense geometry to approximate smooth surfaces

3) Polygons do not displace satisfactorily (due to their essentially discontinuous topology)

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For users of Maya Unlimited, there are often benefits to be gained by converting polygons to subdivision surfaces, especially when displacements are being used. The caveat here is to make sure that the polygonal geometry is not too dense, which could create an inefficient subdivision surface.

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