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Multitype branching processes describe the evolution of populations in which individuals give birth independently according to a probability distribution that depends on their type. In this thesis, we consider the extinction of branching processes with countably infinitely many types. We begin by developing a class of iterative methods to compute the global extinction probability vector $\bm{q}$. In particular, we construct a sequence of truncated and augmented branching processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the infinite vector $\bm{q}$. Besides giving rise to a family of algorithmic methods, our approach leads to new sufficient conditions for $\bm{q}=\bm{1}$ and $\bm{q}<\bm{1}$. When then turn our attention to a specific class of branching processes which we refer to as lower Hessenberg branching processes. These are branching processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j \leq i+1$ only. For this class of processes, we study the set of fixed points of the progeny generating vector. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\bm{q}$ and whose maximum is the partial extinction probability vector $\bm{\tilde{q}}$. We derive a computationally efficient partial extinction criterion. In the case where $\bm{\tilde q}=\bm{1}$, we derive a global extinction criterion, and in the case where $\bm{\tilde{q}}<\bm{1}$, we develop necessary and sufficient conditions for $\bm{q}=\bm{\tilde{q}}$. Finally, we consider a more general class of structured branching processes which we refer to as block lower Hessenberg branching processes. For these processes, we derive partial and global extinction criteria, and we study the probability of extinction $\b