Running MongoDB as a Microservice with Docker and Kubernetes involves using containers to create a lightweight, self-contained sandbox for development and deployment, and orchestration tools to manage the creation, upgrade, and high availability of these containers. To run MongoDB in containers, additional considerations such as persistent storage, IP address stability, and replica set configuration must be taken into account. Implementing a MongoDB replica set using Docker and Kubernetes requires creating multiple pods with services exposing external IP addresses and ports, and utilizing features like Replication Controllers and LoadBalancer services to ensure resiliency and high availability. Running the replica set across multiple clusters and availability zones can provide geographic redundancy and increase reliability.