The aim of this thesis is to derive two distinct model equations for shallow water waves in a continuously stratified fluid and with variable bottom topography. This thesis is divided into two parts. First, we derive a KdV-type shallow water model equation for bi-directional shallow water waves along the equator with a continuous depth-dependent density. Secondly, we derive a KdV-type shallow water model equation for unidirectional shallow water waves along the equator with a continuous depth-dependent density and a bottom that may vary in the direction of wave propagation. We derive both model equations from the governing equations using asymptotic expansion. We obtain model equations that describe the horizontal velocity component for each fixed depth. We derive exact solutions for the second model equation under the assumption that the bottom is slowly varying and perform an analysis of the effect of the varying bottom and the change of density on the waves.