solution, But I think I made some mistakes.

src/App.tsx

@@ -1,7 +1,8 @@
-import { PeoplePage } from './components/PeoplePage';
+// import { PeoplePage } from './components/PeoplePage';
 import { Navbar } from './components/Navbar';
 
 import './App.scss';
+import { Outlet } from 'react-router-dom';
 
 export const App = () => {
   return (
@@ -10,9 +11,7 @@ export const App = () => {
 
       <div className="section">
         <div className="container">
-          <h1 className="title">Home Page</h1>
-          <h1 className="title">Page not found</h1>
-          <PeoplePage />
+          <Outlet />
         </div>
       </div>
     </div>

src/Root.tsx

@@ -0,0 +1,23 @@
+import {
+  Navigate,
+  Route,
+  HashRouter as Router,
+  Routes,
+} from 'react-router-dom';
+import { App } from './App';
+import { PageNotFound } from './components/Pages/PageNotFound';
+import { HomePage } from './components/Pages/HomePage';
+import { PeoplePage } from './components/PeoplePage';
+
+export const Root = () => (
+  <Router>
+    <Routes>
+      <Route path="/" element={<App />}>
+        <Route path="*" element={<PageNotFound />} />
+        <Route path="home" element={<Navigate to="/" replace />} />
+        <Route index element={<HomePage />} />
+        <Route path="people/:peopleSlug?" element={<PeoplePage />} />
+      </Route>
+    </Routes>
+  </Router>
+);

src/components/Navbar.tsx

@@ -1,4 +1,10 @@
+import '../App.scss';
+import { NavLink, useLocation } from 'react-router-dom';
+import classNames from 'classnames';
+
 export const Navbar = () => {
+  const location = useLocation();
+
   return (
     <nav
       data-cy="nav"
@@ -8,17 +14,27 @@ export const Navbar = () => {
     >
       <div className="container">
         <div className="navbar-brand">
-          <a className="navbar-item" href="#/">
+          <NavLink
+            className={({ isActive }) => {
+              return classNames('navbar-item', {
+                'has-background-grey-lighter': isActive,
+              });
+            }}
+            to="/"
+          >
             Home
-          </a>
+          </NavLink>
 
-          <a
-            aria-current="page"
-            className="navbar-item has-background-grey-lighter"
-            href="#/people"
+          <NavLink
+            className={({ isActive }) => {
+              return classNames('navbar-item', {
+                'has-background-grey-lighter': isActive,
+              });
+            }}
+            to={{ pathname: '/people', search: location.search }}
           >
             People
-          </a>
+          </NavLink>
         </div>
       </div>
     </nav>

src/components/Pages/HomePage.tsx

@@ -0,0 +1,5 @@
+import React from 'react';
+
+export const HomePage: React.FC = () => {
+  return <h1 className="title">Home Page</h1>;
+};

src/components/Pages/PageNotFound.tsx

@@ -0,0 +1,5 @@
+import React from 'react';
+
+export const PageNotFound: React.FC = () => {
+  return <h1 className="title">Page not found</h1>;
+};

src/components/PeopleFilters.tsx

@@ -1,16 +1,126 @@
-export const PeopleFilters = () => {
+import React, { useEffect } from 'react';
+import { Person } from '../types';
+import { useSearchParams } from 'react-router-dom';
+import cn from 'classnames';
+
+type Props = {
+  peoples: Person[];
+  onFilter: (persons: Person[]) => void;
+};
+
+export const PeopleFilters: React.FC<Props> = ({ peoples, onFilter }) => {
+  const [serchParams, setSerchParams] = useSearchParams();
+  const query = serchParams.get('query') || '';
+  const centuries = serchParams.getAll('centuries') || [];
+  const sex = serchParams.get('sex') || '';
+
+  function hangeChangeQuery(event: React.ChangeEvent<HTMLInputElement>) {
+    const params = new URLSearchParams(serchParams);
+    const trueQuery = event.target.value.trim();
+
+    if (!trueQuery) {
+      params.delete('query');
+    } else {
+      params.set('query', event.target.value.trim());
+    }
+
+    setSerchParams(params);
+  }
+
+  function hangeChangeSex(sexe: string) {
+    const params = new URLSearchParams(serchParams);
+
+    params.set('sex', sexe);
+    setSerchParams(params);
+  }
+
+  function hangeChangeAges(n: string) {
+    const params = new URLSearchParams(serchParams);
+
+    if (n === 'All') {
+      params.delete('centuries');
+
+      return setSerchParams(params);
+    }
+
+    const newCenturies = centuries.includes(n)
+      ? centuries.filter(centurie => centurie !== n)
+      : [...centuries, n];
+
+    params.delete('centuries');
+    newCenturies.forEach(ages => params.append('centuries', ages));
+
+    return setSerchParams(params);
+  }
+
+  const handleClearAll = () => {
+    const params = new URLSearchParams(serchParams);
+
+    params.delete('centuries');
+    params.delete('query');
+    params.delete('sex');
+    setSerchParams(params);
+
+    return;
+  };
+
+  useEffect(() => {
+    let filtered = [...peoples];
+
+    if (centuries.length > 0) {
+      filtered = filtered.filter(person => {
+        const century = Math.ceil(person.born / 100);
+
+        return centuries.includes(String(century));
+      });
+    }
+
+    if (sex) {
+      filtered = filtered.filter(person => person.sex === sex);
+    }
+
+    if (query.trim()) {
+      filtered = filtered.filter(
+        person =>
+          person.name
+            .toLowerCase()
+            .trim()
+            .includes(query.trim().toLowerCase()) ||
+          person.fatherName
+            ?.toLowerCase()
+            .trim()
+            .includes(query.trim().toLowerCase()) ||
+          person.motherName
+            ?.trim()
+            .toLowerCase()
+            .includes(query.trim().toLowerCase()),
+      );
+    }
+
+    onFilter(filtered);
+  }, [peoples, serchParams, sex, query, onFilter]);
+
   return (
     <nav className="panel">
       <p className="panel-heading">Filters</p>
 
       <p className="panel-tabs" data-cy="SexFilter">
-        <a className="is-active" href="#/people">
+        <a
+          className={cn(sex === '' ? 'is-active' : '')}
+          onClick={() => hangeChangeSex('')}
+        >
           All
         </a>
-        <a className="" href="#/people?sex=m">
+        <a
+          className={cn(sex === 'm' ? 'is-active' : '')}
+          onClick={() => hangeChangeSex('m')}
+        >
           Male
         </a>
-        <a className="" href="#/people?sex=f">
+        <a
+          className={cn(sex === 'f' ? 'is-active' : '')}
+          onClick={() => hangeChangeSex('f')}
+        >
           Female
         </a>
       </p>
@@ -22,6 +132,8 @@
             type="search"
             className="input"
             placeholder="Search"
+            value={query}
+            onChange={event => hangeChangeQuery(event)}
           />
 
           <span className="icon is-left">
@@ -35,40 +147,55 @@
           <div className="level-left">
             <a
               data-cy="century"
-              className="button mr-1"
-              href="#/people?centuries=16"
+              className={cn(
+                'button mr-1',
+                centuries.includes('16') ? 'is-info' : '',
+              )}
+              onClick={() => hangeChangeAges('16')}
             >
               16
             </a>
 
             <a
               data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=17"
+              className={cn(
+                'button mr-1',
+                centuries.includes('17') ? 'is-info' : '',
+              )}
+              onClick={() => hangeChangeAges('17')}
             >
               17
             </a>
 
             <a
               data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=18"
+              className={cn(
+                'button mr-1',
+                centuries.includes('18') ? 'is-info' : '',
+              )}
+              onClick={() => hangeChangeAges('18')}
             >
               18
             </a>
 
             <a
               data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=19"
+              className={cn(
+                'button mr-1',
+                centuries.includes('19') ? 'is-info' : '',
+              )}
+              onClick={() => hangeChangeAges('19')}
             >
               19
             </a>
 
             <a
               data-cy="century"
-              className="button mr-1"
-              href="#/people?centuries=20"
+              className={cn(
+                'button mr-1',
+                centuries.includes('20') ? 'is-info' : '',
+              )}
+              onClick={() => hangeChangeAges('20')}
             >
               20
             </a>
@@ -77,8 +204,11 @@
           <div className="level-right ml-4">
             <a
               data-cy="centuryALL"
-              className="button is-success is-outlined"
-              href="#/people"
+              className={cn(
+                'button is-success',
+                centuries.length !== 0 ? 'is-outlined' : '',
+              )}
+              onClick={() => hangeChangeAges('All')}
             >
               All
             </a>
@@ -87,7 +217,13 @@
       </div>
 
       <div className="panel-block">
-        <a className="button is-link is-outlined is-fullwidth" href="#/people">
+        <a
+          className={cn(
+            'button  is-link is-fullwidth',
+            serchParams.size !== 0 ? 'is-outlined' : '',
+          )}
+          onClick={handleClearAll}
+        >
           Reset all filters
         </a>
       </div>