We investigate lattice structures on locally convex cones, that is, ordered cones that carry a locally convex topology. Examples include the extended reals
\overline{\mathbb{R}}
R
‾
, cones of
\overline{\mathbb{R}}
R
‾
-valued functions and cones of convex subsets of a locally convex vector space. The case of order completeness, where bounded below sets have suprema and infima, is of particular interest. It leads to the notion of order convergence and the introduction of the order topology and its comparison to the given topology of a completely ordered locally convex cone. The use of zero components of a given element allows a more subtle conceptualization of the cancellation law.