My knowledge of automorphic forms basically comes from Lang's $SL_2$ and Borel's automorphic forms on $SL_2$. I try to understand them from representation theory. For example, an integral weight $k$ automorphic form corresponding to the $k$-th character eigenspace for the right $SO_2$ action on $\Gamma\backslash SL_2(\mathbb R)$. In this way, I am not able to understand modular forms of weight $1/2$. Although half integral modular forms have many applications, it is still not clear to me why people only go from integral to half integral, but not more general things such as third integral modular forms.
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