Alternating sign matrices are known to be equinumerous with plane partitions in the totally symmetric self-complementary symmetry class (TSSCPP), but no meaningful bijection is known. In this talk, we discuss various known and new bijections from ASMs or TSSCPPs to other objects, such as webs, order ideals, and pipe dreams, and compare their differences and similarities. We transform TSSCPPs to {0,1,-1}-matrices we call magog matrices, and investigate their enumerative and geometric properties (joint work with Vincent Holmlund). We also discuss a pipe dream theoretic bijection between subsets of ASMs and TSSCPPs in the reduced, 1432-avoiding case (joint work with Daoji Huang). We furthermore discuss joint work with Mathilde Bouvel, Eric Egge, Rebecca Smith, and Justin Troyka studying two notions of pattern avoidance in alternating sign matrices.