Singularity is an obstacle to the treatment of algebraic varieties but at the same time enriches the geometry. Since a terminal threefold singularity is isolated, it is often more flexible to treat it in the analytic category. Artin's algebraisation theorem, Tougeron's implicit function theorem and the Weierstrass preparation theorem are fundamental analytic tools. Taking quotient produces singularities. We clarify the notion of quotient and define the weighted blow-up in the context of which cyclic quotient singularities appear. We furnish a complete classification of terminal threefold singularities due to Reid and Mori. First we deal with singularities of index one and next we describe those of higher index by taking the index-one cover. It turns out that the general member of the anti-canonical system of a terminal threefold singularity is always Du Val. This insight is known as the general elephant conjecture and plays a leading role in the analysis of threefold contractions. Reid established an explicit formula of Riemann-Roch type on a terminal projective threefold. We also discuss canonical threefold singularities and bound the index by means of the above formula.