By a triangulation of a topological space $X$, we mean a simplicial complex $K$ whose geometric carrier is homeomorphic to $X$. The topological properties of the space can be expressed in terms of the combinatorics of its triangulation. Simplicial complexes have gained in prominence after the advent of powerful computers as they are especially suitable for computer processing. In this regard, it is desirable for a triangulation to be as efficient as possible. In this thesis we study different notions of efficiency of triangulations, namely, minimal triangulations, tight triangulations and tight neighborly triangulations.

a) Minimal Triangulations: A triangulation of a space is called minimal if it contains minimum number of vertices among all triangulations of the space. In general, it is hard to construct a minimal triangulation, or to decide if a given triangulation is minimal. In this work, we present examples of minimal triangulations of connected sums of sphere bundles over the circle. b) Tight Triangulations: A simplicial complex (triangulation) is called tight w.r.t field $\mathbb{F}$ if for any induced subcomplex, the induced homology maps from the subcomplex to the whole complex are all injective. We normally take the field to be $\mathbb{Z}_2$. Tight triangulations have several desirable properties. In particular any simplex-wise linear embedding of a tight triangulation (of a PL manifold) is “as convex” as possible. Conjecturally, tight triangulations of manifolds are minimal, and it is known to be the case for most tight triangulations of manifolds. Examples of tight triangulations are extremely rare, and in this thesis we present a construction of an infinite family of tight triangulations, which is only the second of its kind known in literature.

c) Tight Neighborly Triangulations: For dimensions three or more, Novik and Swartz obtained a lower bound on the number of vertices in a triangulation of a manifold, in terms of its first Betti number. Triangulations that meet this bound are called tight neighborly. The examples of tight triangulations constructed in the thesis are also tight neighborly. In addition, it is proved that there is no tight neighborly triangulation of a manifold with first Betti number equal to two.