Volterra integro-ordinary differential equation or system of Volterra integro-ordinary differential equations can be used for mathematical modeling of many advance problems of engineering and sciences like electric circuit problem, the problem of chain reactions and growth problem of cells. Laplace-Carson transform is widely used integral transform and can be used for determining the solution of Volterra integro-ordinary differential equations and their systems. This book is to explore the basic concepts of Volterra integro-ordinary differential equations and Laplace-Carson transforms in a simple, systematic and easy-to-understand manner. The present book is divided into seven chapters which cover all the important topics like introduction of Volterra integro-ordinary differential equations, Laplace-Carson transforms, inverse Laplace-Carson transforms, applications of Laplace-Carson transforms for solving linear Volterra integro-ordinary differential equations (first and second kind) and their systems. Chapter 1 presents introduction of Volterra integro-ordinary differential equations. The definition and examples of integral equation, Volterra integro-ordinary differential equations, primitive of Volterra integro-ordinary differential equations, different types of Volterra integro-ordinary differential equations, linear and non-linear Volterra integro-ordinary differential equations, convolution type Volterra integro-ordinary differential equations and their system are presented in a brief manner. Chapter 2 deals with Laplace-Carson transforms with its fundamental properties. Chapter 3 discusses the inverse Laplace-Carson transforms with its fundamental properties. Chapter 4 discusses the application of Laplace-Carson transform for solving linear Volterra integro-ordinary differential equation of second kind. Chapter 5 discusses the application of Laplace-Carson transform for solving linear Volterra integro-ordinary differential equation of first kind. Chapter 6 deals with the application of Laplace-Carson transform for solving system of linear Volterra integro-ordinary differential equations of second kind. Chapter 7 discusses the application of Laplace-Carson transform for solving system of linear Volterra integro-ordinary differential equations of first kind.