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A linear operator between two Riesz spaces E and F is said to be unbounded order continuous (or uo-continuous, for short) whenever it maps each unbounded order null net in E into an unbounded order null net in F, and it said to be-unbounded order continuous (or uo-continuous, for short) if each unbounded order null sequence in E is mapped into an unbounded order null sequence in F. We begin this talk by a review of some basic notions and results from the theory of Riesz spaces. Then we will recall the unbounded order convergence"(abbreviated, uo-convergence) of nets in Riesz spaces, and demonstrate some recent characterizations of it. Later we will give some properties of uo-continuous and uo-continuous operators. We will also characterize the uo-continuous (respectively, uo-continuous) dual of some well-known Riesz spaces. Finally, as an application of uo-convergence and uo-continuity we establish two variants of Brezis-Lieb lemma in Riesz spaces. PS:This work is a part of ongoing thesis under supervision of Prof. Eduard Emelyanov, Orta Dogu Teknik Universitesi (ODTU).