  For any finite group G, there are several well-established definitions of a
G-equivariant spectrum. In this paper, we develop the definition of a global
orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal
G-spectrum for each finite group G. We use the framework of enriched indexed
categories to make this precise. We also consider equivariant K-theory and
Spin^c-cobordism from this perspective, and we show that the
Atiyah--Bott--Shapiro orientation extends to the global context.
