Algebraic $K$-theory is an invariant of rings that relates to interesting questions in many mathematical subfields including geometric topology, algebraic geometry, and number theory. Although these connections have generated great interest in the study of algebraic $K$-theory, computations are quite difficult. This prompted the development of trace methods for algebraic $K$-theory in which one studies more computable invariants of rings (and their topological analogues) that receive maps from $K$-theory. One such approximation called topological Hochschild homology (THH) has proven foundational to progress in computations of $K$-theory via trace methods. The B{\"o}kstedt spectral sequence is one of the main tools for computing THH. Hesselholt and Madsen developed a $C_2$-equivariant analogue of algebraic $K$-theory for rings with anti-involution called Real algebraic $K$-theory. A Real version of the trace methods story unfolds in this context by studying an approximation of Real $K$-theory called Real topological Hochschild homology (THR). The main result of this thesis is the construction of a Real B{\"o}kstedt spectral sequence which computes the equivariant homology of THR. We then extend our techniques to the case of another equivariant Hochschild theory called $G$-twisted topological Hochschild homology and construct a spectral sequence which computes the $G$-equivariant homology of $H$-twisted THH when $H \leq G$ are finite subgroups of $S^1$. Finally, this thesis explores the algebraic structures present in Real topological Hochschild homology. When the input is commutative, THH has the structure of a Hopf algebra in the homotopy category. Work of Angeltveit-Rognes further shows that this structure lifts to the B{\"o}kstedt spectral sequence. We show in this thesis that when the input is commutative, THR has a Hopf algebroid structure in the $C_2$-equivariant stable homotopy category.
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