  We introduce and develop a theory of limits for sequences of sparse graphs
based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the $L^p$ theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.
