Abstract: The notion of a one-dimensional, a two-dimensional, or a three-dimensional geometric object is fairly intuitive. A natural way to think of dimensions makes it relatively easy to define, though not necessarily easy to visualise, geometric objects also in four or five dimensions or indeed in n dimension for any positive integer n. It is however much harder to make sense of the notion of a geometric object with a non-integer (fractal) dimension, such as 1.5 for example, and to imagine what such an object might look like. In this introductory talk I will give an elementary alternative definition of the dimension of a geometric object and show how this definition extends easily to fractal dimensions. I will also give some simple examples of geometric objects with fractal dimensions. I will discuss the way in which the geometry of many (in fact, most) objects in nature can arguably be modelled by geometric shapes with fractal dimension rather than the regular shapes familiar in classical Euclidean geometry. Finally, I will describe how objects with fractal dimensions occur naturally in certain areas of mathematics such as Dynamical Systems.