Preheat the oven to 350°F (175°C). Grease a 9×5 inch loaf pan.
In a large bowl, combine the mashed bananas, granulated sugar, and melted butter. Mix well.
Add the eggs one at a time, beating well after each addition.
Stir in the vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the banana mixture, stirring until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove from oven and let cool in the pan for 10 minutes.
Transfer the banana bread to a wire rack to cool completely before slicing.