Beam tracing combines the flexibility of ray tracing and the speed of polygon rasterization. However, beam tracing so far only handles linear transformations; thus, it is only applicable to linear effects such as planar mirror reflections but not to non-linear effects such as curved mirror reflection, refraction, caustics and shadows. In this paper, we introduce non-linear beam tracing to render these non-linear effects. Non-linear beam tracing is highly challenging because commodity graphics hardware supports only linear vertex transformation and triangle rasterization.We overcome this difficulty by designing a non-linear graphics pipeline and implementing it on top of a commodity GPU. This allows beams to be non-linear where rays within the same beam do not have to be parallel or intersect at a single point. Using these non-linear beams, real-time GPU applications can render secondary rays via polygon streaming similar to how they render primary rays. A major strength of this methodology is that it naturally supports fully dynamic scenes without the need to pre-store a scene database. Utilizing our approach, non-linear ray tracing effects can be rendered in real-time on a commodity GPU under a unified framework.