In this paper we are concerned with definably, with or without parameters, complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o-minimal if and only if all their definable subsets are finite disjoint unions of definably connected subsets. We pay attention to how simply a definable gap in an expansion is so. (...) Next we prove that over parametrically definably complete expansions of ordered fields, all one-to-one definable continuous functions are monotone and open. Moreover, in both parameter and parameter-free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine-Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets. (shrink)