Modern concepts of the mechanisms of protein aggregation with an emphasis on immunoglobulin-like domains (Ig domains) as a structural platform predisposed to the formation of aggregates with amyloid properties are presented in this review. Particular attention is paid to the muscle proteins titin and myosin-binding protein C, which form not fibrillar, but amorphous amyloid aggregates with a cross-β structure without an increase in the total content of the secondary β-structure, capable of binding thioflavin T (ThT) and Congo red. The absence of a nucleation phase during the formation of amyloid aggregates of these proteins, as well as their partial disaggregation, allows us to declare a new, previously undescribed pathway of amyloid protein aggregation. We refer to it as self-templating amorphous β-assembly (conformational conversion by a prion-like mechanism), characteristic of multidomain proteins of the sarcomeric cytoskeleton.