How to add custom font weights to Material UI?

I am using the same behavior, with numbers, tested with all the 100/200/300/400/500/600/700 and it worked as well:

import { MuiThemeProvider, createMuiTheme } from '@material-ui/core/styles';
import { Typography } from '@material-ui/core/';

const THEME = createMuiTheme({
  typography: {
    "fontFamily": "\"MyCustomFont\"",
    "fontSize": 20,
    "lineHeight": 1.5,
    "letterSpacing": 0.32,
    useNextVariants: true,
    suppressDeprecationWarnings: true,
    h6: {
      "fontWeight": 600,
    },
  },
});

<MuiThemeProvider theme={THEME}>
  <Typography variant="h6" color="inherit" align="center" style={{ margin: "1rem" }}>
      "Some text"
  </Typography>
</MuiThemeProvider>

The Box element can help. Ignore the Typography element.

See the code example below

<Box fontWeight="fontWeightLight">…
<Box fontWeight="fontWeightRegular">…
<Box fontWeight="fontWeightMedium">…
<Box fontWeight={500}>…
<Box fontWeight="fontWeightBold">…

See official docs for more details