Two approvals, in order
Planning. The City's Planning Division reviews the house against Malibu's zoning and its Local Coastal Program. For a new single-family home that review typically runs as a Coastal Development Permit, often with companion requests: Site Plan Review for height above 18 feet, a minor modification for a setback, or neighborhood standards for size. Planning approval ends with a green stamp on the plans.
Building Safety. With the green stamp, the project moves to plan check in the Building Safety Division of the Community Development Department, through the City's online Development Portal. The City routes the drawings to its internal reviewers. Outside agencies are the applicant's job, and the City names them: the Los Angeles County Fire Department and Los Angeles County Water District 29 among them. I manage those submittals with your design team so the building permit is not waiting on a letter nobody requested.
One rule shapes every Malibu schedule. A Coastal Development Permit does not take effect until all appeals, including appeals to the California Coastal Commission, are resolved, and it expires if construction has not started within three years of final City action unless it is extended for due cause.