Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan and set aside.
In a large mixing bowl, combine the mashed bananas, granulated sugar, and melted butter. Mix well until smooth.
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, salt, ground cinnamon, ground nutmeg, and ground allspice.
Gradually add the dry ingredients to the banana mixture, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and spread evenly.
Bake in the preheated oven for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove the banana bread from the oven and let it cool in the pan for 10 minutes.
Transfer the bread to a wire rack and allow it to cool completely before slicing and serving.