Extend the theories of Lagrangian Floer cohomology, obstructions to $HF^{*},$ and Fukaya categories $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include Lagrangians $L$ in $M$ with isolated conical singularities modelled on stable special Lagrangian cones $C$ in ${\mathbin{\mathbb{C}}}^{m},$ such as the $T^{2}$-cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ in (2.4). The main technical issues will involve studying moduli spaces of $J$-holomorphic discs $\Sigma$ in $M$ whose boundaries $\partial\Sigma$ lie in $L$ and pass through singular points of $L$.