The thesis is devoted to the weighted criteria for integral operators with positive ker- nels in variable exponent Lebesgue and amalgam spaces. Similar results for multiple kernel operators defined with respect to a Borel measure in the classical Lebesgue spaces are also obtained. More precisely, we established necessary and sufficient conditions on a weight function v governing the boundedness/compactness of the weighted positive kernel operator Kv f (x) = v(x) x 0 k(x, y)f (y)dy from Lp(·) (R+ ) to Lq(·) (R+ ) under the local log-H ̈lder continuity condition and the decay condition at o infinity on the exponents p and q. In the case when Kv is bounded but not compact, two-sided estimates of the measure of non-compactness (essential norm) for Kv are obtained in terms of the weight v and kernel k. Criteria guaranteeing the boundedness /compactness of weighted kernel operators defined on R+ (resp. on R) in variable ex- ponent amalgam spaces are found. The kernel operators under consideration involve, x for example, the Riemann-Liouville transform Rα f (x) = 0 f (t) dt, (x−t)1−α 0 < α < 1. Necessary and sufficient conditions ensuring weighted estimates for maximal and po- tential operators in variable exponent amalgam spaces are also established under the local log-H ̈lder continuity condition on exponent of spaces. Further, we establish o criteria on measures governing the boundedness of integral operators with product positive kernels defined with respect to a Borel measure in the classical Lebesgue spaces. Finally, we point out that Fefferman-Stein type inequality for the multi- ple Riemann-Liouville transform defined with respect to a product Borel measure is derived.