This thesis deals with analytic multivalent and meromorphic functions defined on an open unit disc E and punctured open unit disc E∗. Some basics definitions and earlier results are stated to illustrate the concepts and important classes of Geometric Function Theory. Then, new class of p-valent meromorphic functions which maps the punctured open unit disc onto certain domains bounded by conic curves in w-plane are studied. This class of functions is defined by associating it with hypergeometric functions. Some properties for certain class of p-valent meromorphic functions like inclusion, closure and coefficient estimate are given. Further, two classes of p-valent meromorphic functions on conic domain are defined using subordination. It is found that such functions display certain geometric properties, such as closure properties under certain integral operators and inclusion properties with the coefficient bounds. Further, certain classes of analytic functions of Janowski type associated to k-symmetric and (j,i)-symmetric points are characterised and their geometric properties are investigated. Some properties like inclusion relations, closure theorems and radii of certain classes of starlike and convex meromorphic functions are given. The (j,i)-symmetric functions for the conic domains are also discussed. Finally, a class of analytic functions associated with generalised S˘ al˘agean operator is introduced and some properties are discussed. These include the radius problem ,coefficient estimate and convolution. Certain class of analytic functions defined by a linear operator is also studied. Some properties such as inclusion and closure are obtained as well.