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We study relations between the additive and the multiplicative groups of atwo-sided skew brace. In particular, we prove that if the additive group of atwo-sided skew brace is finite solvable (respectively, finitely generatednilpotent, finitely generated residually nilpotent, finitely generatedresidually finite), then the multiplicative group of this skew brace issolvable (respectively, solvable, residually solvable, residually finite).Also, we prove that if the multiplicative group of a two-sided skew brace isnilpotent of nilpotency class $k$, then the additive group of this skew braceis solvable of class at most $2k$. The letter result generalizes the result ofByott which says that if the multiplicative group of a finite skew brace isabelian, then the additive group of this skew brace is solvable.  In addition, we solve two problems (Problem 19.49 and Problem 19.90(a))concerning skew braces which are formulated in the Kourovka notebook.

Connections between properties of the additive and the multiplicative groups of a two-sided skew brace