Abstract: The sequence of numbers 1,2,7,42, 429, ... (http://oeis.org/A005130) counts several remarkable combinatorial objects, one of which is alternating sign matrices (ASMs). These matrices have become an object of fervent study by combinatorialists because of the inherent simplicity and beauty of enumerative results related to them which are notoriously difficult to prove. I will give a brief history of how these objects were introduced and eventually enumerated, which will already lead to connections with other areas (most notably lattice models in statistical physics). In the second part, I will speak about recent joint work with Ilse Fischer that gives several new refined enumeration results for symmetry classes of ASMs. The talk will be self-contained.