Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if forevery neighbourhood V of the identity e in G, there exists a natural number n such that G = V ⁿ (resp.we have G = ∪ _n∈N V ⁿ ). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded)premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudoboundedparatopological group such that G is a Lusin space, then is G a topological group. We present some examplesof paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroupwhich is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.