I've heard that there are many convenient categories for algebraic topology. Such categories often have many nice properties like being cartesian closed, complete, cocomplete, the forgetful functor creates limits, containing all "nice" spaces like CW complexes and topological manifolds, et cetera. But the only convenient category for algebraic topologies that I know are the category of compact generated weakly Hausdorff (CGWH) spaces. Can someone briefly summarize other such categories and their advantages and disadvantages when compared to the category of CGWH spaces?
From the nLab (although I was the author of these words):
A reasonably large class of examples, including the examples of compactly generated spaces and sequential spaces, is given in the article by Escardó, Lawson, and Simpson (ref). These may be outlined as follows. An exponentiable space in $Top$ is a space $X$ such that $X \times : Top \to Top$ has a right adjoint. These may be described concretely as corecompact spaces (spaces whose topology is a [[continuous lattice]]). Suppose given a collection $\mathcal{C}$ of corecompact spaces, with the property that the product of any two spaces in $\mathcal{C}$ is a colimit in $Top$ of spaces in $\mathcal{C}$. Such a collection $\mathcal{C}$ is called productive. Spaces which are $Top$colimits of spaces in $\mathcal{C}$ are called $\mathcal{C}$generated.
Theorem (Escardó, Lawson, Simpson): If $\mathcal{C}$ is a productive class, then the full subcategory of $Top$ whose objects are $\mathcal{C}$generated is a coreflective subcategory of $Top$ (hence complete and cocomplete) that is cartesian closed.
Other convenience conditions, such as inclusion of CWcomplexes and closure under closed subspaces, are in practice usually satisfied as well. For example, if closed subspaces of objects of $\mathcal{C}$ are $\mathcal{C}$generated, then closed subspaces of $\mathcal{C}$generated spaces are also $\mathcal{C}$generated. If the unit interval $I$ is $\mathcal{C}$generated, then so are all CWcomplexes.
A number of examples are scattered throughout the paper.

2$\begingroup$ See also Booth, P.I. and Tillotson, J. Monoidal closed categories and convenient categories of topological spaces Pacific J. Math. {88} (1980) 3353. $\endgroup$ – Ronnie Brown Dec 6 '18 at 12:39

$\begingroup$ Freelyavailable pdf link for Booth–Tillotson: msp.org/pjm/1980/881/pjmv88n1p03s.pdf $\endgroup$ – David Roberts Dec 7 '18 at 20:33
If you want a convenient category of pointed spaces, then the category $\mathbf{NG}_0$ of pointed numerically generated spaces, discussed for instance in
Kazuhisa Shimakawa, Kohei Yoshida, Tadayuki Haraguchi, Homology and cohomology via enriched bifunctors Kyushu Journal of Mathematics, 2018, Volume 72, Issue 2, Pages 239252, doi:10.2206/kyushujm.72.239
is one. The authors say that numerically generated is equivalent to being $\Delta$generated, as discussed in Dan Dugger's Notes on Deltagenerated spaces.

$\begingroup$ An important advantage of $\Delta$generated spaces  shared with sequential spaces, or any full subcategory of $Top$ which is the colimitclosure of a small subcategory (a la Todd's response) is that they are locally presentable, which simplifies certain infinitary constructions like the small object argument. $\endgroup$ – Tim Campion Dec 7 '18 at 18:15
One property that one sometimes wants, but which is not satisfied by most of the "usual" convenient categories, is that of being locally cartesian closed, or equivalently that each pullback functor $f^* : \mathcal{C}/Y \to \mathcal{C}/X$ has a right adjoint $f_*$.
However, I don't know of any nontrivial subcategory of the usual category $\mathrm{Top}$ of topological spaces that is locally cartesian closed. The closest locally cartesian closed categories to $\mathrm{Top}$ that I know of are quasitoposes, such as the category of subsequential spaces (which contains the category of sequential spaces mentioned by Todd, and is locally presentable) or pseudotopological spaces (which contains the entire category $\mathrm{Top}$, but is not locally presentable).
Another approach, used by May and Sigurdsson, is to mix two convenient categories to approximate local cartesian closure. As explained in their book, if $\mathcal{K}$ denotes the category of notnecessarilyweakHausdorff $k$spaces, then the pullback functor $f^*:\mathcal{K}/Y\to \mathcal{K}/X$ has a right adjoint as long as $X$ and $Y$ are weak Hausdorff.