It has been seen that semi-classical string solutions play a major role in the study of the string/gauge correspondence. Giant magons are a particular set of these solutions, which are intimately related to sine-Gordon solitons. Considered to be the basic solitons of classical strings in the $AdS_5 \times S^5$ background, giant magnons are the building blocks one can use to understand other string solutions. The theory of classical strings in both $AdS_5 \times S^5$ and $AdS_4 \times CP^3$ backgrounds is integrable, and the problem of finding the spectrum of solutions can be described by a Riemann-Hilbert problem in the so called Algebraic Curve formalism. This talk will focus on the use of the Algebraic Curve formalism to study the spectrum of these string solitons, first in the simpler case of $AdS_5 \times S^5$, and then in the much richer $AdS_4 \times CP^3$ case.