  Topological surgery in dimension $3$ is intrinsically connected with the
classification of $3$-manifolds and with patterns of natural phenomena. In this
expository paper, we present two different approaches for understanding and
visualizing the process of $3$-dimensional surgery. In the first approach, we
view the process in terms of its effect on the fundamental group. Namely, we
present how $3$-dimensional surgery alters the fundamental group of the initial
manifold and present ways to calculate the fundamental group of the resulting
manifold. We also point out how the fundamental group can detect the
topological complexity of non-trivial embeddings that produce knotting. The
second approach can only be applied for standard embeddings. For such cases, we
give new visualizations for both types of $3$-dimensional surgery as different
rotations of the decompactified $2$-sphere. Each rotation produces a different
decomposition of the $3$-sphere which corresponds to a different visualization
of the $4$-dimensional process of $3$-dimensional surgery.

Geometric Topology