General Relativity, in spite of being the most successful gravitational theory has left some problems without answer. Brane-Gravity makes an attempt to answer some of these problems. According to the theory of brane-gravity, major portion of gravity lies in higher-dimensional "bulk" space-time as well as very small portion of gravity and standard model particles are trapped to the 3-brane. Here, starting from an overview of extra dimensions, the text discusses essentials of Brane-Gravity and Brane Cosmology. Followed by this introduction, the text investigates certain aspects of braneworld modifications to cosmological dynamics. In particular, it is devoted to following two important aspects of Brane Cosmology (i) Acceleration and deceleration of the universe (ii) Universe as a thermodynamical system and validity of thermodynamical laws. Various issues associated with these two aspects have been discussed along with extensive literature survey. Brane-Gravity and its applications to cosmology require deep understanding of mathematics and physics. Keeping this in mind, the detail calculation are explained in appendices so that text is useful to young research scholars.